Quartic Exercises

A correspondence between quartic étale algebras over a field and quadratic étale extensions of cubic étale algebras is set up and investigated. The basic constructions are laid out in general for sets with a profinite group action and for torsors, and translated in terms of étale algebras and Galois algebras. A parametrization of cyclic quartic algebras is given.

V = {I, (1, 2) (3,4), (1,3) (2,4), (1,4)(2, 3)}, which is the kernel of the action of S 4 on its three Sylow 2-subgroups.Numbering from 1 to 3 these Sylow subgroups, we get an exact sequence of groups (1) 1 Let F be an arbitrary field and P ∈ F [X] be a separable polynomial of degree 4. Let also F s be a separable closure of F and Q ⊂ F s be the subfield generated by the roots of P .The Galois group Gal(Q/F ) can be viewed as a subgroup of S 4 through its action on the roots of P .The subfield L of Q fixed under Gal(Q/F ) ∩ V is generated by the roots of a cubic resolvent, as was shown by Lagrange.For a given quartic polynomial P , there are actually many polynomials of degree 3 which qualify as cubic resolvents; only the extension L/F is an invariant of P (or of Q).
Galois cohomology provides another viewpoint on this construction.Since S n is the automorphism group of the étale F -algebra F n = F × • • • × F , it is well-known that the Galois cohomology set H 1 (F, S n ) is in canonical one-to-one correspondence with the isomorphism classes of étale F -algebras of degree n, see [3, (29.9)].The map ρ in (1) induces a map which associates to every quartic étale F -algebra Q a cubic étale F -algebra R(Q) uniquely determined up to isomorphism.If P ∈ F [X] is a separable polynomial of degree 4 with cubic resolvent R, and if Q is the factor algebra Q = F [X]/P , then R(Q) F [X]/R, see §4. 3.
Our first aim is to make explicit the construction of R(Q) from Q.But this construction can be further extended.Each of the three Sylow 2-subgroups of S 4 contains two transpositions, and each transposition is in one and only one Sylow subgroup, hence the set of transpositions can be viewed as a double covering of the set of Sylow 2-subgroups.Therefore, the conjugation action of S 4 on its six transpositions defines a map λ : S 4 → S 2 S 3 , where the wreath product S 2 S 3 is viewed as the group of automorphisms of a double covering of a set of three elements (see §3.1).The map λ extends to an isomorphism of groups λ : S 2 × S 4 ∼ → S 2 S 3 , see §4.2.The set H 1 (F, S 2 S 3 ) classifies the quadratic étale extensions of cubic étale F -algebras (see §3.2), and the induced bijection λ1 : associates to every pair consisting of a quartic étale F -algebra Q and a quadratic étale F -algebra a quadratic étale extension of the cubic resolvent R(Q).In §4.3, we give an explicit construction of this quadratic extension, and we describe in §4.4 the inverse of λ1 , attaching a quartic étale F -algebra and a quadratic étale F -algebra to any quadratic étale extension of a cubic étale F -algebra.
In the final sections, we classify quartic étale algebras and their associated quadratic extensions of cubic algebras according to their decomposition into direct products of fields (see §5.3) and we parametrize cyclic quartic extensions.

References 49
La recherche des extensions d'un corps k dont le groupe de Galois sur k est S 4 ou A 4 n'est pas autre chose, du point de vue des algébristes du XIX e siècle, que la théorie de l'équation du 4 e degré.C'est un problème pour lequel ces algébristes n'avaient que du mépris.(A.Weil) 1. Γ-sets and coverings 1.1.Basic constructions on Γ-sets.Let Γ be a profinite group, which will be fixed throughout this section.Finite sets with a continuous action of Γ are called Γ-sets.We let |X| denote the number of elements in a Γ-set X.If X is a Γ-set with n elements, and k is a positive integer, k ≤ n, we let Σ k (X) denote the set of k-tuples of pairwise distinct elements of X and Λ k (X) the set of k-element subsets of X; thus The action of Γ on X induces actions on Σ k (X) and Λ k (X), hence Σ k (X) and Λ k (X) are Γ-sets, and we have The symmetric group S k acts on Σ k (X) by permutation of the entries, and we may consider Λ k (X) as the set of orbits of Σ k (X) under this action, i.e. as the quotient Γ-set Λ k (X) = Σ k (X)/S k .For k = n, we may also consider the action of the alternating group A n on Σ n (X).The quotient is called the discriminant of X and denoted by ∆(X), ∆(X) = Σ n (X)/A n , see [3, p. 291].This is a Γ-set with |∆(X)| = 2 if n ≥ 2.
If n is even, n = 2m, let be the map which associates to every m-element subset of X its complementary subset.Since γ 2 X = Id, this map defines an action of S 2 on Λ m (X).The map γ X is Γ-equivariant (i.e.compatible with the action of Γ), hence the quotient R(X) = Λ m (X)/S 2 is a Γ-set.It is the set of partitions of X into m-element subsets.
The following observations are clear: Proposition 1.2.Let X, X , be Γ-sets of two elements.
(a) The Γ-action on X * X is trivial.
(b) If the Γ-action on X is trivial, then X * X X. (Note that the isomorphism is not canonical.)(c) The operation * defines a group structure on the set of isomorphism classes of Γ-sets of two elements.
See §3.2 for a cohomological interpretation of the group structure induced by * .
which is a double covering.In particular, for any Γ-set X of two elements and any covering Y Proof.Recall from (2) the map and, similarly, Therefore, the element We thus have a canonical map then ψ(ω) = ψ(ω ), hence ψ is onto.On the other hand, if ω is obtained from ω by an even number of changes as above, then ψ(ω) = ψ(ω ).Therefore, , and it follows that ψ factors through the map This completes the proof.
For later use, we record another case where the map δ of (2) can be used in the computation of a discriminant.Let X be a Γ-set of two elements.For any Γ-set Y , we may consider the double covering Y π2 ←− X × Y given by the projection.
Proposition 1.8.Let X be a Γ-set of two elements, and let Y π ← − Z be a double covering.There is a canonical isomorphism Proof.For simplicity of notation, let X = {+, −} and denote by the canonical automorphism γ Z/Y .We may then identify X * Z with the set of formal polynomials ζ − ζ, for ζ ∈ Z.Note however that the Γ-action on these polynomials is not linear, since Γ may act non trivially on {+, −}.The structure map On the other hand,

Étale algebras and extensions
In this section, F is an arbitrary field.We denote by F s a separable closure of F and let Γ = Gal( Conversely, starting from a Γ-set X with |X| = n, we may let Γ act by semi-linear automorphisms on the F s -algebra of maps Map(X, F s ).The fixed F -algebra (see [3, (18.19)]), so that the functors M and X define an anti-equivalence between the category É t F of étale F -algebras (with F -algebra homomorphisms) and the category Set Γ of Γ-sets.Under this anti-equivalence, the direct product (resp.tensor product) of F -algebras corresponds to the disjoint union (resp.direct product) of Γ-sets: for E 1 , E 2 étale F -algebras, there are obvious identifications Moreover, if G is a group acting on an étale F -algebra E by F -automorphisms, then for the fixed subalgebra E G we have since E G is the equalizer of the automorphisms σ : E → E for σ ∈ G, and X(E)/G is the co-equalizer of the corresponding automorphisms of X(E).Through the antiequivalence É t F ≡ Set Γ , the constructions on Γ-sets defined in §1.1 therefore have counterparts in the category of étale F -algebras.The aim of this section is to make them explicit.
2.1.Basic constructions on étale algebras.Let E be an étale F -algebra of dimension n.Under the canonical isomorphism E M X(E) , the idempotents of E correspond to the characteristic functions of Γ-subsets of X(E).If e ∈ E is the characteristic function of a subset Y ⊂ X(E), then multiplication by e defines an isomorphism E/(1 − e)E ∼ → eE.Moreover, X(eE) = Y and under the anti-equivalence É t F ≡ Set Γ , the map E → eE corresponds to the inclusion X(E) ← Y .
Example 2.1.If E is the split étale F -algebra E = F n , then X(E) is in duality with the set e 1 , . . ., e n of minimal idempotents of E, namely X(E) = {ξ 1 , . . ., ξ n } where The idempotent corresponding to a subset Let E be an arbitrary étale F -algebra of dimension n.For any integer k with 1 ≤ k ≤ n, we let s k ∈ E ⊗k be the idempotent corresponding to the characteristic function of the subset In particular, for k = 2 the idempotent 1 − s 2 is the characteristic function of the diagonal of X(E) × X(E) = X(E ⊗ E).It is the separability idempotent of E, see [3, p. 285].For k ≥ 3, the idempotent s k can also be defined in terms of the separability idempotent of E, see [8, p. 42], [3, p. 320].
The symmetric group S k acts on E ⊗k by permutation of the factors, and the idempotent s k is fixed under this action, so S k also acts on Σ k (E).We consider the fixed subalgebra We have since under the anti-equivalence É t F ≡ Set Γ the fixed algebra under S k corresponds to the factor set under the S k -action.
The discriminant of E is defined by and we have For an arbitrary étale F -algebra E of dimension n, the algebra Λ k (E) can also be viewed as an algebra of linear transformations of the exterior power k E (where E is just regarded as a vector space), as we now show.
Multiplication on the left defines an F -algebra homomorphism (the regular representation) As pointed out by Saltman [7, Lemma 1.1], the image of (E ⊗k ) S k in End F (E ⊗k ) preserves the kernel of the canonical map E ⊗k → k E. Therefore, there is an induced F -algebra homomorphism Lemma e i1 e j1 ∧ . . .∧ e i k e j k = e j1 ∧ . . .∧ e j k .
In view of the lemma, the homomorphism (3) induces an F -algebra homomorphism Saltman [7,Lemma 1.3] has shown that the image of this map has dimension For instance, for a, x 1 , . . ., Now, consider the case where n is even, n = 2m.Since dim n E = 1, and the exterior product m E × m E → n E is a nonsingular bilinear pairing, there is an adjoint involution γ on End F ( m E), defined by the equation Proposition We may now define Proof.It suffices to see that under the anti-equivalence É t F ≡ Set Γ , the automorphism γ E of Λ m (E) corresponds to the permutation γ X(E) of X Λ m (E) = Λ m X(E) .Again, we may extend scalars to a separable closure of F and assume E is split.Using the same notation as in the preceding proof, we may identify X(E) with the dual basis of e 1 , . . ., e n .Equation (4) shows that γ E (e {i1,...,im} ) = e {k1,...,km} , where {k 1 , . . ., k m } is the complementary subset of {i 1 , . . ., i m } in {1, . . ., n}, and the proof is complete.
When dim E = 2, the algebra E is called a quadratic étale F -algebra.In the notation above, we then have m = 1, so Λ m (E) = E, hence E carries a canonical involutive automorphism γ E .Let E, E be two quadratic étale F -algebras, with canonical involutive automorphisms γ E , γ E .The tensor product γ E ⊗ γ E defines a S 2 -action on E ⊗ F E , and we let Proof.
Let Quad(F ) be the set of isomorphism classes of quadratic étale F -algebras, which is in bijection under X with the set of isomorphism classes of Γ-sets of two elements.The following analogue of Proposition 1.2 is easily proved, either directly or by reduction to Proposition 1.2 under the anti-equivalence É t F ≡ Set Γ : Proposition 2.7.Let E, E be two quadratic étale F -algebras.2.2.Extensions of étale algebras.An étale F -algebra B containing an F -algebra A (necessarily étale) is called an extension of degree d of A if it is a free A-module of rank d.Equivalently, this condition means that the inclusion Proposition 2.8.There is a canonical surjective map Λ n (B) → Ω(B/A), and Proof.The set X Ω(B/A) is the set of F -algebra homomorphisms (B ⊗n ) Sn → F s which map i ⊗n (s A n ) to 1. Since X (B ⊗n ) Sn = X(B) n /S n , every such homomorphism is the orbit of an n-tuple (ξ 1 , . . ., ξ n ) of elements of X(B); the condition that the homomorphism maps i ⊗n (s A n ) to 1 is equivalent to the following: the homomorphism (A ⊗n ) Sn → F s associated to the n-tuple i * (ξ 1 ), . . ., i * (ξ n ) maps s A n to 1.In view of the definition of s A n , this means that i * (ξ 1 ), . . ., i * (ξ n ) are pairwise distinct, hence Of course, this condition implies that ξ 1 , . . ., ξ n are pairwise distinct, hence {ξ 1 , . . ., ξ n } ∈ Λ n X(B) .Thus, The inclusion Ω X(B)/ X(A) ⊂ Λ n X(B) yields the canonical surjective map Λ n (B) → Ω(B/A) under the anti-equivalence É t F ≡ Set Γ .Now, suppose d = 2, so that dim F B = 2n.The canonical involutive automorphism γ X(B)/ X(A) on X(B) corresponds to a canonical involutive automorphism γ B/A of B such that On the other hand, there is also a "complementary subset" map Since this map preserves Ω X(B)/ X(A) , the corresponding map γ B : Λ n (B) → Λ n (B) induces an involutive automorphism on Ω(B/A), which we also denote by γ B , and we may consider the subalgebra of fixed points By definition, it is clear that Example 2.9.Suppose A and B are split of dimensions 3 and 6 respectively, with minimal idempotents e 1 , e 2 , e 3 and f 1 , As observed in Example 2.2, and {e σ(1) ⊗ e σ(2) ⊗ e σ(3) | σ ∈ S 3 } is the set of minimal idempotents of Σ 3 (A).Denoting in general by u ⊗ v ⊗ w the sum of the six products obtained by permuting the factors u, v, w (so where u 1 = u, u 2 = v, u 3 = w), the minimal idempotents of Ω(B/A) are The involution γ B interchanges g i and g i for i = 0, . . ., 3, hence the minimal idempotents of S(B/A) are Let B, B be quadratic extensions of an étale F -algebra A. The canonical map The following result is clear: Let Quad(A) be the set of isomorphism classes over A of quadratic extensions of A. The following proposition is the analogue of Proposition 1.5: Proposition 2.11.Let E be a split étale F -algebra of dimension 2 (i.e.E F ×F ), and let B/A be a quadratic extension of étale F -algebras.

(a) The extension (B *
It is clear that Propositions 1.4, 1.6 and 1.8 have analogues for étale algebras.We record them below.Proposition 2.12.Let E be an étale F -algebra of dimension 2, and let B/A and B /A be quadratic extensions of an étale F -algebra A. There are canonical isomorphisms:

Cohomology of permutation groups
3.1.Permutations.For any finite set X, let S X be the symmetric group of X, i.e. the group of all permutations of X.Thus, S X = S n for X = {1, . . ., n}.Every permutation of a set X of n elements induces a permutation of the sets Σ k (X), Λ k (X) (for k ≤ n), ∆(X), and of R(X) if n is even.There are therefore canonical group homomorphisms be the group of automorphisms of the covering.The map (σ, τ ) → τ identifies S Z/Y to a subgroup of S Z .On the other hand, the map (σ, τ ) → σ defines a surjective homomorphism β Z/Y : S Z/Y → S Y whose kernel is isomorphic to S n d upon identifying each fiber of π with {1, . . ., d}.Therefore, the group S Z/Y has order (d!) n n! and can be identified to a wreath product For later use, note that the kernel of s Z/Y is the "diagonal" subgroup S 2 of S Z/Y , whose nontrivial element is γ Z/Y .This diagonal subgroup is central in S Z/Y .
On the other hand, every permutation of a set X with n = 2m elements induces an automorphism of the covering R(X) Moreover, the composition of λ X and the canonical homomorphism The proof is left to the reader.

3.2.
Cohomology and Γ-sets.As in §1, we denote by Γ a profinite group, which will be fixed throughout this subsection.The action of Γ on a set X with |X| = n can be viewed as a group homomorphism Γ → S X S n .
Since the isomorphism S X S n depends on the indexing of the elements in X, the homomorphism Γ → S n is defined by X up to conjugation by an element in S n .Therefore, there is a canonical one-to-one correspondence between isomorphism classes of Γ-sets of n elements and the cohomology set H 1 (Γ, S n ) (with the trivial action of Γ on S n ), by definition of this cohomology set.Under this correspondence, the distinguished element of H 1 (Γ, S n ) is mapped to the set with trivial Γ-action.Since the symmetric group S 2 is abelian, there is an abelian group structure on H 1 (Γ, S 2 ).We leave it to the reader to verify that the product of the isomorphism classes of the Γ-sets X, X with Similarly, every covering and there is a canonical one-to-one correspondence between isomorphism classes of coverings of degree d of Γ-sets of n elements and the cohomology set H 1 (Γ, S d S n ), which maps the distinguished element of the cohomology set to the covering with trivial Γ-action.The basic constructions in subsections 1.1 and 1.2 yield canonical maps of cohomology sets through the induced homomorphisms of permutation groups (see §3.1).For instance, if X is a Γ-set of n elements and k ≤ n, the canonical homomorphism σ k : S X → S Σ k (X) induces a morphism of pointed sets ).Since the Γ-action on Σ k (X) is induced by the Γ-action on X through σ k , the morphism σ 1 k maps the isomorphism class of X to the isomorphism class of Σ k (X).A similar statement obviously holds for the morphisms Let S Z/Y be the group S Z/Y with this action of Γ, and define S Y similarly.By [3, (28.8)], there are canonical bijections The exact sequence of Γ-groups yields an exact sequence in cohomology The kernel of β 1 Z/Y is the set of isomorphism classes of coverings of degree d of Y .By [3, (28.4)], this kernel is in canonical bijection with the orbit space of H 1 (Γ, T Z/Y ) under the fixed-point group H 0 (Γ, S Y ).Note that H 0 (Γ, S Y ) is the group of permutations of Y which commute with the action of Γ; in other words, it is the group of automorphisms of the Γ-set Y , This group acts naturally on C d (Y ), and When the Γ-action on Y is transitive, let Γ 0 ⊂ Γ be the stabilizer of an arbitrary (but fixed) element of Y , so that Y Γ/Γ 0 .Then we may identify T Z/Y with Map(Γ/Γ 0 , S d ) and get a canonical bijection in the spirit of Shapiro's lemma see [3, (28.20)].
Whatever the action of Γ on Y , when d = 2 the group Note that this operation is generally not defined on the orbit set ker 3.3.Torsors.As in the preceding subsection, we fix a profinite group Γ.Besides the correspondence between H 1 (Γ, S n ) and the isomorphism classes of Γ-sets of n elements explained in the preceding subsection, there is also a one-to-one correspondence between H 1 (Γ, S n ) and isomorphism classes of S n -torsors, i.e. of Γ-sets of n! = |S n | elements with a free action of S n (on the right) compatible with the Γ-action (on the left), see [3, (28.14)].Combining the correspondences, we obtain a bijection between isomorphism classes of Γ-sets of n elements and S n -torsors.To the isomorphism class of the Γ-set X with |X| = n corresponds the class of Σ n (X), which clearly is an S n -torsor.Conversely, we associate to an S n -torsor Σ the class of Σ/S n−1 .The n projections are Γ-equivariant maps and satisfy where Σ is an S n -torsor and A similar construction can be given for coverings, since the set 3.4.Cohomology and étale algebras.In this section, F is an arbitrary field, F s is a separable closure of F and Γ = Gal(F s /F ) is the absolute Galois group of F .The anti-equivalence É t F ≡ Set Γ induces a canonical bijection between the set of isomorphism classes of étale F -algebras of dimension n and the set of isomorphism classes of Γ-sets of n elements.Since the latter set is in one-to-one correspondence with the cohomology set H 1 (Γ, S n ) (see §3.2), there is also a canonical bijection between H 1 (Γ, S n ) and isomorphism classes of étale F -algebras of dimension n.This bijection can be set up directly, by identifying S n with the group of automorphisms of the split algebra F n s .More precisely, given an étale algebra A and an isomorphism α : Conversely, given a cocycle (f γ ) γ∈Γ in S n , the corresponding étale algebra is = x} where Γ acts on F n s entrywise.As in §3.2, the basic constructions on étale algebras of §2.1 can be interpreted in terms of morphisms of cohomology sets.Details are left to the reader, as well as the analogues for extensions of étale algebras and the cohomology of wreath products.We simply note for later use the canonical isomorphism where Quad(F ) is the group of isomorphism classes of quadratic étale F -algebras (see Proposition 2.7).For any étale F -algebra A, we also have canonical isomorphisms The operation * A is generally not defined on this set.

Galois algebras.
As in the preceding subsection, F is an arbitrary field and Γ is the absolute Galois group of F .Let G be a finite group.A G-Galois F -algebra is an étale F -algebra of dimension |G| with an action of G by F -algebra automorphisms such that the algebra of fixed points is F (see [3, (18.15)]).Equivalently, an étale F -algebra E of dimension |G| with an action of G is G-Galois if and only if the Γ-set X(E) is a G-torsor for the induced action of G. Therefore, the discussion of torsors in §3.3 has an analogue in terms of Galois algebras, and the set H 1 (Γ, S n ) is also in one-to-one correspondence with the set of isomorphism classes of S n -Galois F -algebras.
If E is an étale F -algebra of dimension n, the algebra Σ n (E) has a natural action of S n , for which it is an S n -Galois algebra.There are n embeddings ε i : E → Σ n (E) corresponding to the projections π i : Σ n X(E) → X(E).They are defined explicitly as follows: for x ∈ E, where Every S n -Galois closure of E is isomorphic to (Σ n (E), ε n ).This construction was suggested by Saltman, see [8, p. 42].
We next sketch an analogue of the Galois closure for extensions of étale algebras, on the model of the corresponding construction for coverings in §3.3.
Let B/A be an extension of degree d of an étale F -algebra A of degree n.Viewing B as an étale A-algebra of degree d, we have an where s

There are d canonical embeddings ε
The algebra Σ(B/A) is an extension of Σ n (A) of degree (d !) n and there exist nd canonical embeddings ( 5) commutative for all i and j.We say that the extension Σ(B/A)/Σ n (A) is an and Example 3.6.Suppose, as in Example 2.9, that A and B are split of dimensions 3 and 6 respectively, with minimal idempotents e 1 , e 2 , e 3 and f 1 , The algebra Σ(B/A) is split.Its 48 minimal idempotents are where σ varies in S 3 .The action of S 3 on these idempotents is clear, and the fixed subalgebra is Ω(B/A) as described in Example 2.9.
Proposition 3.7.Let B/A be an extension of étale algebras of degree d and let Proof.Let Σ be the subalgebra of Σ(B/A) generated over Σ(A) by the ε ij (b).We show that Σ = Σ(B/A).We may assume that A and B are split and we assume for simplicity that n = 3 and d = 2.We use the notations of Example 2.9.Let Since b generates B, the 6 elements β i , β j are pairwise different.We have

The symmetric group on four elements
In the rest of this paper, we focus on various aspects of étale algebras of dimension 4 (called quartic étale algebras) which, as explained in the preceding sections, can be viewed from the perspective of Γ-sets of 4 elements, or of the cohomology of S 4 , or of S 4 -torsors, or of S 4 -Galois algebras.It turns out that there is a group isomorphism which relates the various "quartic" notions listed above to those associated with the cohomology of S 2 S 3 : quadratic extensions of cubic étale algebras, double coverings of sets of 3 elements, S 2 S 3 -torsors and S 2 S 3 -Galois algebras.We explain this relation in the simplest case, namely Γ-sets and coverings, and then give the cohomological viewpoint in the next subsection.In the last two subsections, we give explicit constructions of R(Q) for a quartic algebra Q, making clear that this algebra is related to the resolvent cubic of quartic equations, and of Ω(B/A) and S(B/A) for a quadratic extension of a cubic algebra A.

4.1.
Sets of four elements and double coverings.In this subsection, Γ is an arbitrary profinite group.Suppose X is a Γ-set with |X| = 4, as in Example 1.1, where the constructions of Λ 2 (X) and R(X) are made explicit.Our first observation concerns the discriminants of Λ 2 (X) and R(X): Moreover, the Γ-action on ∆ Λ 2 (X) is trivial.
The proof is a straightforward verification.To see that the Γ-action on ∆ Λ 2 (X) is trivial, it suffices to observe that every transposition on X -hence every permutation of X-induces an even permutation of Λ 2 (X).For another approach, see Proposition 4.7.
To get a better grasp of the various constructions associated with X, it is useful to think of X as the set of diagonals of a cube. 1 Each pair of diagonals determines a diagonal plane (passing through an edge and its opposite), hence Λ 2 (X) is identified with the set of diagonal planes of the cube.The map γ X carries each diagonal plane to the plane through parallel edges, and R(X) can therefore be identified with the set of directions of the edges.The canonical map R(X) ε ← − Λ 2 (X) maps each diagonal plane to the direction of the edges it contains.The set Ω Λ 2 (X)/ R(X) consists of (unordered) triples of diagonal planes with different edge directions.For each such triple τ , either the intersection of the planes is a diagonal, or the intersection is just the center of the cube.However, if the intersection is a diagonal, then the intersection of the complementary triple τ = γ Λ2(X)/ R(X) (τ ) is the center.Therefore, we may associate to the pair {τ, τ } ∈ S Λ 2 (X)/ R(X) a unique diagonal in X, and obtain a map Proposition 4.2.For |X| = 4, the map Proof.From the definition, it is clear that Φ is Γ-equivariant.Bijectivity of Φ is checked by direct inspection.
To put this result into perspective, consider the full subcategory Set 4 Γ of Set Γ whose objects are the Γ-sets of four elements, and the category Cov As we now show, we may however define a canonical bijection (We denote both coverings by ε.) Our first goal is to give a geometrical interpretation of the set ∆(Z).Recall the map δ Z : Ω(Z/Y ) → ∆(Z) of (2).By Proposition 1.4, this map is onto.It may therefore be used to consider ∆(Z) as a quotient of Ω(Z/Y ), the set of vertices of the cube.It is easily checked that the four vertices which have the same image under δ Z are the vertices of a regular tetrahedron whose edges are the diagonals of the faces of the cube.Therefore, we may identify ∆(Z) with the set {T 1 , T 2 } of such tetrahedra.Given a diagonal plane λ ∈ Λ 2 S(Z/Y ) and a tetrahedron T ∈ ∆(Z), there is a unique face z ∈ Z whose intersection with λ is an edge of T .The same face z intersects the "complementary" plane λ following an edge of the "complementary" tetrahedron T .Therefore, the map (T, λ) → z induces a well-defined map Remark. 2 For X, X ∈ Set 4 Γ , every morphism of coverings f : Λ 2 (X)/ R(X) → Λ 2 (X )/ R(X ) induces a morphism S Λ 2 (X)/ R(X) → S Λ 2 (X )/ R(X ) , hence, by Proposition 4.2, a morphism f : X → X .The functor Λ carries f to f , hence it is full.Since S • Λ is equivalent to the identity, the functor Λ is also faithful.Moreover, Corollary 4.4 shows that every covering Z/Y ∈ Cov 2 3  Γ such that the Γaction on ∆(Z) is trivial is isomorphic to a covering of the form Λ(X). Therefore, it follows from [6, Theorem 1, p. 93] that Λ is an equivalence of categories from Set 4  Γ to the full subcategory of Cov 2 3 Γ whose objects are the coverings Z/Y with trivial Γ-action on ∆(Z).
In order to take into account the double coverings of Γ-sets of three elements which have non-trivial action on the discriminant, we consider the product category Set 2 Γ × Set 4 Γ whose objects are pairs (U, X) of Γ-sets with |U | = 2 and |X| = 4, and extend Λ and S to functors by Propositions 1.6 and 1.7.The Γ-action on ∆ Λ 2 (X) is trivial by Proposition 4.1, hence the right-most Γ-set in ( 6) is isomorphic to U by Proposition 1.2(b).Note that the latter isomorphism is not canonical, hence Ŝ • Λ is not naturally equivalent to the identity on Set 2 Γ × Set 4 Γ .However, since Ŝ • Λ(U, X) (U, X), we have an isomorphism between sets of isomorphism classes.An alternative proof in cohomology can be derived from Diagram (10).Theorems 4.5 and 4.6 have analogues in terms of quartic étale algebras and double coverings of cubic algebras, whose statements are left to the reader.
Remark.The functor Ŝ is faithful since Λ • Ŝ is equivalent to the identity.Moreover, every (U, X) ∈ Set 2 Γ × Set 4 Γ is isomorphic to an object of the form Ŝ(Z/Y ) (namely, Z/Y = Λ(U, X)).Furthermore, every morphism f : ), hence by Proposition 4.3 a morphism f : Z/Y → Z /Y .We may check that f = Ŝ( f ), hence the functor Ŝ is full.By [6, Theorem 1, p. 93], it defines an equivalence of categories Cohomology.This subsection presents the cohomological perspective on Theorem 4.6.We use the same notation as in the preceding subsection.
The isomorphisms λ and ŝ can also be described in purely group-theoretical terms.The subgroup S 2 = S 2 × {1} ⊂ S 2 × S 4 is mapped to the "diagonal" subgroup S 2 ⊂ S 2 S 3 , which is the center of S 2 S 3 .On the other hand, the restriction of λ to S 4 = {1} × S 4 is a homomorphism λ : S 4 → S 2 S 3 which may be described as the action of S 4 by conjugation on its transpositions.Indeed, S 4 contains six transpositions, which sit by pairs in the three Sylow 2subgroups of S 4 .The map which carries each transposition to the unique Sylow 2-subgroup which contains it is a double covering of a set of three elements.Note that the composition of λ and the canonical homomorphism β : S 2 S 3 → S 3 is the surjective homomorphism ρ : S 4 → S 3 which is the action of S 4 on its three Sylow 2-subgroups.(Alternately, the map ρ may be identified with the canonical homomorphism S X → S R(X) for X = {1, 2, 3, 4}, since there is a canonical one-to-one correspondence between R(X) and the Sylow 2-subgroups of S X .)The kernel of ρ is the Vierergruppe V.
By definition, it is clear that the first component of ŝ is the signature map sgn : since the map S Z/Y → S ∆(Z) is the signature.The second component is a homomorphism which is the action of S 2 S 3 on its four Sylow 3-subgroups.(There is a natural one-to-one correspondence between the Sylow 3-subgroups of S Z/Y and S(Z/Y ).) The image of λ is the kernel of sgn, by Proposition 4.1 or by Proposition 3.1, and the map s splits λ (if the Sylow 3-subgroups of S 2 S 3 are suitably indexed).The maps ρ, λ and β, and the inclusions ι, η, are part of the following commutative diagram with exact rows and columns: where σ is the sum.Since the exact sequences in this diagram are split, there is a corresponding commutative diagram of exact sequences in cohomology: (10) Cohomology yields an alternative proof of Proposition 4.1: Proposition 4.7.For any Γ-set X with |X| = 4, ∆(X) ∆ R(X) .
Proof.The commutative diagram The first part of the proposition follows since ρ 1 maps the isomorphism class of X to the isomorphism class of R(X), and sgn 1 maps the isomorphism class of any Γ-set to the isomorphism class of its discriminant.The second part follows from the fact that is a zero-sequence.
As another application of cohomology, we describe the quartic étale algebras which have a given resolvent cubic.
Let R be a Γ-set of three elements, and let X 0 = R {0} be the Γ-set of four elements obtained by adjoining to R a fixed point 0. To each partition of X 0 into 2-element subsets, we may associate the unique element r ∈ R such that {0, r} is in the partition, and thus identify As in §3.2, we let Γ act by conjugation on the groups S X0 , S R , and denote by S X0 , S R the Γ-groups thus defined.The inclusion R → X 0 yields a Γ-equivariant embedding S R → S X0 which splits the map ρ : and the isomorphism classes of X ∈ Set 4 Γ such that R(X) R are in one-toone correspondence with ker ρ 1 = im ι 1 .They form a pointed set with the isomorphism class of X 0 as distinguished element.Note that exactness of the sequence (11) does not mean that ι 1 is injective.In fact, the group Aut Γ (R) = H 0 (Γ, S R ) acts on H 1 (Γ, V X0 ), and im ι 1 is in canonical bijection with the orbit set H 1 (Γ, V X0 )/ Aut Γ (R), by [3, (28.4)].
To give a more explicit description, we use a variant of Diagram (10).First, observe that we may identify Λ 2 (X 0 ) to {1, 2} × R, as follows: we map a 2-element subset U ⊂ X 0 to (1, r) if 0 / ∈ U and r / ∈ U , to (2, r) if U = {0, r}.We may then identify the double covering Λ 2 (X 0 )/ R(X 0 ) to Let Z 0 = {1, 2} × R. As above, we let Γ act by conjugation on S Z0/R , and denote by S Z0/R the corresponding Γ-group.As in §3.2, let T Z0/R be the kernel of the canonical map β Z0/R : S Z0/R → S R .The exact sequence is split, and induces an exact sequence in cohomology As above, there is a canonical bijection The orbit set on the right side may therefore be identified with the set of isomorphism classes of double coverings of R, see §3.2.Consider the commutative diagram analogous to (10), ( 12) 1.The left vertical sequence is an exact sequence of groups.It shows that H 1 (Γ, V X0 ) can be identified with the kernel of σ 1 .Recall from §3.2 that H 1 (Γ, T Z0/R ) is in canonical bijection with the set C 2 (R) of isomorphism classes over R of double coverings of R, and that H 1 (Γ, S 2 ) classifies Γ-sets of two elements up to isomorphism.By commutativity of the lower square in (12), the map σ 1 carries every double covering to the isomorphism class of its discriminant.Therefore, we may identify H 1 (Γ, V X0 ) with the group C 2 0 (R) of isomorphism classes over R of double coverings of R with trivial discriminant, We have thus shown: Proposition 4.8.The set of isomorphism classes of sets X of four elements such that R(X) R is in canonical bijection with the set C 2 0 (R)/ Aut Γ (R) of isomorphism classes of double coverings of R with trivial discriminant.
Suppose now Γ is the absolute Galois group of a field F with separable closure F s , and let A be a cubic étale F -algebra.Using the anti-equivalence Set Γ ≡ É t F , we may translate Proposition 4.8 into the following statement, where we denote by Quad 0 (A) the set of isomorphism classes over A of quadratic extensions of A whose discriminant (as F -algebra) is trivial: Proposition 4.9.The set of isomorphism classes of quartic étale F -algebras Q with R(Q) A is in canonical bijection with the set Quad 0 (A)/ Aut F (A) of Fisomorphism classes of quadratic extensions of A with trivial discriminant.
The next proposition gives an explicit description of the group Quad 0 (A).Proposition 4.10.Let N 1 (A) be the (multiplicative) group of elements of A of norm 1 and let T 0 (A) be the (additive) group of elements of A of trace 0. (a) Proof.If A is a field, the action of Γ on X(A) is transitive.Letting Γ 0 ⊂ Γ be the absolute Galois group of a copy of A in F s , we have Quad(A) H 1 (Γ 0 , S 2 ) as observed in §3.2, and the map σ 1 can be interpreted as the corestriction under which the corestriction corresponds to a map induced by the norm.Its kernel is If char F = 2, we identify S 2 with {0, 1} ⊂ F s .The exact sequence under which the corestriction corresponds to a map induced by the trace.Its kernel is If A is not a field, it decomposes into a direct product of fields, In the first case, and the map σ 1 can be again interpreted as induced by the norm or the trace.We may then use the same arguments as above.The case where A F × F × F is left to the reader.
Following §3.5, the set H 1 (Γ, S R ) for R = X(A) is also in one-to-one correspondence with the set of isomorphism classes of S 3 -Galois F -algebras, where the distinguished element corresponds to the isomorphism class of the S 3 -Galois closure Σ 3 (A).Likewise, the set H 1 (Γ, S X0 ) classifies S 4 -Galois F -algebras up to isomorphism, with the class of Σ 4 (F × A) as distinguished element.The upper exact sequence of Diagram (12) shows that the S 4 -Galois F -algebras M which are the S 4 -Galois closure of an étale quartic F -algebra Q with R(Q) A are in oneto-one correspondence with Quad 0 (A)/ Aut F (A).Using Proposition 4.10, we may make this correspondence explicit as follows: Proposition 4.11.Let A be a cubic étale F -algebra, identified with a subalgebra of its S 3 -Galois closure Σ 3 (A), and let ρ ∈ S 3 be an element of order 3.
(a) If char F = 2, let a ∈ A × be such that N A/F (a) = 1, and set (b) If char F = 2, let a ∈ A be such that T A/F (a) = 0, and set In each case, there is a S 4 -action on M which endows it with the structure of an S 4 -Galois algebra.The quartic subalgebra Q = M S3 satisfies R(Q) A. Moreover, every S 4 -Galois F -algebra which is the S 4 -Galois closure of a quartic étale F -algebra Q with R(Q) A is of this form.
Remark.Similar constructions are described by Serre (see [10] 3 ) and by Weil (for the construction of dyadic field extensions with Galois group S 4 , see [13,Section 31]).4.3.Quartic étale algebras.In this subsection, our goal is to make explicit the relation between resolvent cubics of quartic polynomials and the construction of R(Q) for Q a quartic étale F -algebra.Our first observation is a direct consequence of Proposition 4.1 (see also Proposition 4.7).Proposition 4.12.Let Q be a quartic étale F -algebra.There is a canonical isomorphism ∆ R(Q) Proof.The proposition readily follows from Proposition 4.1 under the anti-equivalence É t F ≡ Set Γ , since the ∆, R and Λ 2 construction commute with the functor X.
Recall from [12] that "the" resolvent cubic of a quartic polynomial ( 13) with roots u 1 , u 2 , u 3 , u 4 in an algebraic closure, is the polynomial g(v) with roots This polynomial has the form ( 14) where An alternative resolvent cubic suggested by Lagrange [4, (32), p. 266] in characteristic different from 2 has roots Since w i = α 2 1 − 4v i for i = 1, 2, 3, this polynomial has the form where Now, let Q be a quartic étale algebra over a field F of arbitrary characteristic.
Proposition 4.13.Suppose x ∈ Q is a generating element with minimal polynomial f (u) as in (13), so that the coefficient α 1 is the trace as in (14).
Proof.Extending scalars, we may assume that Q is split, with a basis (e 1 , e 2 , e 3 , e 4 ) consisting of minimal (orthogonal) idempotents.Then R(Q) is split and e 1 ⊗ e 2 + e 2 ⊗ e 1 + e 3 ⊗ e 4 + e 4 ⊗ e 3 , e 1 ⊗ e 3 + e 3 ⊗ e 1 + e 2 ⊗ e 4 + e 4 ⊗ e 2 and e 1 ⊗ e 4 + e 4 ⊗ e 1 + e 2 ⊗ e 3 + e 3 ⊗ e 2 is a basis of R(Q) consisting of minimal idempotents.Let x = x 1 e 1 + x 2 e 2 + x 3 e 3 + x 4 e 4 with x 1 , x 2 , x 3 , x 4 ∈ F .Since x generates Q, the coefficients x i are pairwise distinct.Computation shows that where {i, j, i , j } = {1, 2, 3, 4}.It follows that This shows that γ Q (λ x )λ x is a root of a polynomial g whose roots in F are These roots are distinct since an easy computation yields Therefore, γ Q (λ x )λ x is a generator of R(Q) and g is its minimal polynomial.Similarly, and the same arguments show that with minimal polynomial h as in ( 16) if the characteristic of F is different from 2. Since the condition κ 3 = 0 implies that the elements , the minimal polynomial of λ x over R(Q) is as stated in the proposition.
Remark.Allison gives in [1, §6] another description of the algebra R(Q), for Q a quartic étale F -algebra.For x ∈ Q, he considers the image Assuming that the characteristic of F is different from 2, Allison defines R(Q) as the span of the products f x • f y , for x, y ∈ Q of trace zero.This definition coincides with the definition in §2.1 under an isomorphism induced by ϕ 2 .
4.4.Quadratic extensions of cubic étale algebras.Let A be an étale Falgebra of dimension 3, and let B be an extension of degree 2 of A. In the same spirit as the preceding subsection, we proceed to give explicit equations for generating elements of S(B/A).
Our first observation is the analogue of Proposition 1.4 through the anti-equivalence between coverings of Γ-sets and extensions of étale F -algebras.Proposition 4.14.There is a canonical embedding ∆(B) → Ω(B/A) such that In the case where B (and therefore A) is split, the image of ∆(B) in Ω(B/A) is spanned by the idempotents in the notation of Example 2.9.
In the general case, for b ∈ B we set for brevity b = γ B/A (b), and  Proof.It suffices to prove the assertions after scalar extension.We may therefore assume B is split, and use the same notation as in Example 2.9.Let Computation yields We have δ b ∈ F if and only if the coefficients of this condition holds if and only if δ b = δ b .The equivalence of (a) and (b) is thus proved.
To complete the proof, let This element generates Ω(B/A) over ∆(B) if and only if u 0 , . . ., u 3 are pairwise distinct and u 0 , . . ., u 3 are pairwise distinct.Computation yields Therefore, ω b generates Ω(B/A) over ∆(B) if and only if ( this proves the equivalence of (a') and (b').
Recall from [3, p. xviii] the forms T = T A/F , S = S A/F and N = N A/F of degrees 1, 2 and 3 respectively on A, such that the generic polynomial of every element a ∈ A has the form (The form T is the trace, and N is the norm.)For a ∈ A, let a i = ε i (A) ∈ Σ 3 (A).One has T (a) = a 1 + a 2 + a 3 , S(a) = a 1 a 2 + a 1 a 3 + a 2 a 3 and N (a) = a 1 a 2 a 3 .
Fix b ∈ B, and let

Computation yields
Proposition 4.16.If ω b generates Ω(B/A) over ∆(B), its minimal polynomial is Use that in the split case, the four roots of the minimal polynomial of ω b over ∆(A) are (with the notations of the proof of Proposition 4.15) the elements u i , i = 0, . . ., 3.

If char
and Moreover, the minimal polynomial of δ b over F is and the minimal polynomial of ω b over ∆(B) is Proof.Proposition 4.15 shows that δ b generates ∆(B) and that ω b generates Ω(B/A) over ∆(B).This last fact can also be seen directly: the algebra Ω(B/A) is generated over F by the elementary symmetric functions in the b i ; since  In contrast with Proposition 4.17, ω b does not generate Ω(B/A) since (b − b) 2 = 1 does not generate A (see Proposition 4.15).One could take for example ω ab as a generator of Ω(B/A), since (ab − ab) 2 = a 2 (assuming that a 2 also generates A, which for a cubic étale algebra is in general the case).However a simpler minimal polynomial is obtained for the element (with the notations of Equation 19).Moreover we have hence the proof is complete.

Special actions on four elements
As in the preceding sections, Γ denotes a profinite group.The constructions on Γ-sets given in §1.1 take a special form when the Γ-action has some particular properties.For instance, if the action on a set X is not transitive, then the orbits X 1 , . . ., X r under Γ yield a Γ-set decomposition X = X 1 . . .X r .
Even if the Γ-action on X is transitive, the induced action on Σ n (X) may not be transitive.
Proposition 5.1.Let X be a Γ-set with |X| = n, and let α : Γ → S X be the action of Γ.If S X : α(Γ) = r, there is a Γ-set decomposition and the subgroups G i are conjugate in S n .Moreover, the Γ-sets Ω 1 , . . ., Ω r are isomorphic.
Conjugation by σ maps G j to G i and the action of σ defines an isomorphism of Γ-sets Ω i ∼ − → Ω j .
Note that if the Γ-action on X is transitive, then the map X πn ← − − Σ n (X) restricts to each Ω i to define a covering X ← Ω i .Extending Definition 3.2, this covering may be regarded as a G i -Galois closure of X.
Taking for Γ the absolute Galois group of a field F , and using the anti-equivalence É t F ≡ Set Γ of Section 2, we may adapt the construction above to étale algebras.Disjoint unions of Γ-sets correspond to direct product decompositions of algebras, hence an étale F -algebra is a field if and only if the Γ-action on X(E) is transitive.
If dim E = n, Proposition 5.1 thus yields a direct product decomposition of the S n -Galois closure Σ n (E) into isomorphic fields Each L i is a Galois extension of F with Galois group G i ⊂ S n isomorphic to the image of the action Γ → S X(E) .If E is a field, each L i can be regarded as a Galois closure of E/F , see [3, (18.22)].
In the rest of this section, we consider the particular case where n = 4.To determine the various possibilities for the image of the action Γ → S 4 , we list the subgroups of S 4 .Proposition 5.2.In the symmetric group S 4 , • the alternating group A 4 is the unique subgroup of order 12; • there are four subgroups of order 6; they are conjugate to S 3 ; • there are three subgroups of order 8; they are pairwise conjugate and isomorphic to the dihedral group D 4 .Moreover, every proper subgroup of S 4 is contained in at least one of the subgroups listed above.
Proof.Any subgroup of index 2 in S 4 must contain all the Sylow 3-subgroups of S 4 .Since these Sylow subgroups are generated by the cycles of length 3, the first claim is clear.
A subgroup of order 6 in S 4 cannot be transitive on {1, 2, 3, 4}.On the other hand, it has an orbit of 3 elements since it contains a Sylow 3-subgroup, hence it must be the isotropy group of one of 1, 2, 3, or 4.
The dihedral group D 4 acts on the four vertices of a square, hence it may be considered as a subgroup of S 4 .It is then identified with a 2-Sylow subgroup of S 4 , and all the 2-Sylow subgroups are conjugate.
Finally, let G ⊂ S 4 be a subgroup.If its order is divisible by 3, then it is 3, 6 or 12, hence G is contained in A 4 or in a conjugate of S 3 .If its order is a power of 2, then G is contained in a 2-Sylow subgroup.
In the following subsections, we examine the additional information on a Γ-set X with |X| = 4 (or on a quartic étale F -algebra Q) when the Γ-action factors through a subgroup S 3 or D 4 .We then collect the information to obtain a classification of quartic étale F -algebras in §5.3.
5.1.Action through S 3 .Suppose first the action of Γ leaves an element x ∈ X invariant.It then preserves a disjoint union decomposition where |R| = 3.The 2-element subsets of X containing x are in one-to-one correspondence with R, hence Λ 2 (X) decomposes as Moreover, the "complementary subset" involution γ X on Λ 2 (X) interchanges R and Λ 2 (R) and defines an isomorphism R Λ 2 (R).Therefore, we have canonical isomorphisms Λ 2 (X) R R and R (X) R.
(See also §4.2.)Assuming Γ is the absolute Galois group of a field F , we may translate the results above in the framework of étale F -algebras.
Proposition 5.3.Let Q be a quartic étale F -algebra.If the Γ-action on X(Q) factors through a subgroup S 3 ⊂ S X(Q) , then there is a cubic field extension L/F such that Moreover, the following conditions are equivalent: The following proposition establishes the existence of a "dual" Γ-set X: Proposition 5.5.If the equivalent conditions of Proposition 5.4 hold, then there exists a Γ-set X ∈ Set 4 Γ , with Γ-action through a Sylow 2-subgroup of S X , such that Moreover, X is a double covering of ∆( X), and X is a double covering of ∆(X).
If the Γ-action on ∆(X) is not trivial, the Γ-set X is canonically determined.If the Γ-actions on ∆(X) and ∆( X) are not trivial, then there is a canonical isomorphism X = X.
Proof.Let D ∈ R(X) be a fixed point of Γ. Define X as the complementary subset in Λ 2 (X) of the fiber ε −1 (D) under the canonical map R(X) . The set X is thus canonically determined if Γ has a unique fixed point D ∈ R(X), or, equivalently by Proposition 5.4, if the Γ-action on ∆(X) is not trivial.
We proceed to prove that X satisfies the stated properties.To clarify the discussion, we use geometric language.If D = {x 1 , x 2 }, {x 3 , x 4 } , we identify X with the set of vertices of a square, letting {x 1 , x 2 } and {x 3 , x 4 } be the pairs of opposite vertices.We may thus identify D to the set of diagonals of the square, and we have a decomposition where X is the set of pairs of adjacent vertices, which may be identified with the set of edges of the square.(Note that X may also be viewed as the dual square of X in the sense of polytope theory.)There is a "dual" decomposition where X is identified with the set of pairs of adjacent edges (by mapping every such pair to their common vertex) and M is the set of pairs of parallel edges, which may be identified with the medians of the square.The "complementary subset" involutions γ X and γ X preserve the decompositions (23) and (24), and the set of orbits of X (resp.X) under γ X (resp.γ X ) can be identified with M (resp.D), hence If the Γ-action on ∆( X) is not trivial, then M is the unique fixed point of R( X), and X ⊂ Λ 2 ( X) is the complementary subset of the fiber of M under the canonical map R( X) ← Λ 2 ( X), hence X = X.This completes the proof.
We may use the Γ-set X to obtain information on the Γ-action on X, as follows: Proposition 5.6.Let X ∈ Set 4 Γ be a Γ-set satisfying the equivalent properties of Proposition 5.4, and suppose the Γ-action on ∆(X) is not trivial, so the dual set X is uniquely determined.The following properties are equivalent: (a) the Γ-action on X factors through a cyclic subgroup C 4 ; (b) X X; (c) ∆( X) ∆(X).
Proof.If the action of Γ factors through C 4 , we may regard X as the set of vertices of an oriented square, and use the orientation to define a canonical isomorphism X ∼ − → X, proving (a)⇒(b).Since the implication (b)⇒(c) is clear, it only remains to prove (c)⇒(a).If the image of Γ under the action contains the Vierergruppe V X , then there is an element in Γ which acts trivially on M and non-trivially on D, hence ∆(X) ∆( X).Similarly, if some element of Γ acts by a single transposition on X, then it acts trivially on D and non-trivially on M , hence ∆(X) ∆( X).Therefore, (c) implies that the image of the action of Γ contains at most cycles of length 4 and one element of V X .
Remark.The Γ-set D * M = ∆(X) * ∆( X) can be identified with the set of orientations of the square.
For the following proposition, recall that the dihedral group D 4 contains two non-conjugate elementary abelian subgroups C 2 × C 2 .Viewing D 4 as a subgroup of S 4 , one of these subgroups is V (= D 4 ∩ A 4 ).The other one is generated by two disjoint transpositions; it is not transitive on {1, 2, 3, 4}.
Proposition 5.7.Let X ∈ Set 4 Γ be a Γ-set satisfying the equivalent properties of Proposition 5.4, and suppose the Γ-action on ∆(X) is not trivial, so the dual set X is uniquely determined.The following properties are equivalent: (a) the Γ-action on X factors through an elementary abelian subgroup C 2 ×C 2 = V X ; (b) the Γ-action on X factors through V X ; (c) the Γ-action on ∆( X) is trivial.
The proof is left to the reader.
Finally, we consider the case where the Γ-action on X factors through V X .
Proposition 5.8.For a Γ-set X ∈ Set 4 Γ , the following conditions are equivalent: (a) the Γ-action on X factors through the Vierergruppe V X ; (b) the Γ-action on R(X) is trivial; (c) the Γ-set Λ 2 (X) has a decomposition into 2-element subsets stable under the canonical involution of Λ 2 (X)/ R(X), (d) X satisfies the equivalent conditions of Proposition 5.4 and Γ acts trivially on ∆(X).
Moreover, if these conditions hold, then the Γ-action on D 1 * D 2 * D 3 is trivial.
Proof.The Vierergruppe can be defined as the subgroup of S X which leaves invariant all the partitions of X into 2-element subsets, hence Taking for Γ the absolute Galois group of a field F , we may translate in terms of étale F -algebras the results of this subsection, by using the anti-equivalence É t F ≡ Set Γ of §2.By a quartic 2-algebra we mean an étale algebra which is a quadratic extension of a quadratic étale algebra.These algebras can be characterized through Proposition 5.4: Proposition 5.9.For a quartic F -algebra Q, the following conditions are equivalent: (a) the Γ-action on X Proposition 5.5 proves for every quartic 2-algebra Q the existence of a "dual" quartic 2-algebra Q, which is canonically determined if ∆(Q) is not split.This algebra is a quadratic extension of ∆(Q), and Q is a quadratic extension of ∆( Q).Moreover, Q and Q satisfy the following relations: We record a few special cases: Proposition 5.10.Let Q be a quartic 2-algebra over F . ( ( ( where K 1 and K 2 are quadratic field extensions of F , then one may take These results are easily derived from Propositions 5.6, 5.7 and 5.8.Note that split quadratic algebras are allowed in (2), and that the case where Q may use the computations of §4.3 to give an explicit description of Q. Proposition 5.11.Let Q be a quartic 2-algebra over F , and let K ⊂ Q be a quadratic étale F -algebra.Denote by the canonical involution of K over F .Proposition 4.13 also shows that the projection λx of λ x onto Q satisfies (29) λ2 x + T (y) λx + w = 0.If ℘ −1 (y) and ℘ −1 (y) are determined in such a way that ℘ −1 (y) + ℘ −1 (y) = ℘ −1 (T (y)), computation shows that ℘ −1 (T (y)) T (y) + ℘ −1 (y)℘ −1 (y) also satisfies (29), hence we may identify Q with F ℘ −1 (y)℘ −1 (y) .We refer to [5] for a description of quartic 2-extensions of fields in characteristic different from 2.

Classification of quartic algebras.
Combining the results of § §5.1 and 5.2, we obtain a classification of quartic étale F -algebras Q based on the action of the absolute Galois group Γ of F on X(Q).We summarize the various possibilities for is the image of the Γ-action.The letters N and L are used for sextic and cubic separable field extensions of F , and K, Ǩ, K 1 , K 2 for quadratic separable field extensions of F .A quartic separable field extension is called an S 4 -quartic (resp.A 4 -quartic) if its Galois closure has Galois group isomorphic to S 4 (resp.A 4 ).
S 4 -quartic K L S 3 -cubic N ⊃ L

Cyclic quartic algebras
Let F be an arbitrary field with absolute Galois group Γ = Gal(F s /F ).Quartic étale F -algebras Q such that the Γ-action on X(Q) factors through a cyclic group C 4 can be endowed with the structure of a C 4 -Galois algebra.(In the table of §5.3, they can be found in the lines α(Γ) = {1}, α(Γ) = C 2 ⊂ V and α(Γ) = C 4 .)Fixing a generator of C 4 (or, equivalently, choosing an isomorphism C 4 Z/4Z), we may consider a C 4 -Galois F -algebra as a pair (Q, ν) where Q is a quartic étale F -algebra and ν is an F -algebra automorphism of Q such that If C 4 is embedded in S 4 , the corresponding map in cohomology H 1 (Γ, C 4 ) → H 1 (Γ, S 4 ) maps the isomorphism class of (Q, ν) to the isomorphism class of Q.Since C 4 is an abelian group, the set H 1 (Γ, C 4 ) is an abelian group.The group structure on Cycl 4 (F ) is induced by the following composition law (see [2]): The class of the split algebra F 4 with the cyclic permutation of factors is the neutral element.The squaring map ρ : C 4 → S 2 fits into an exact sequence Since H 1 (Γ, S 2 ) Quad(F ) (see §3.4), the induced exact sequence in cohomology takes the form (30) 1 → Quad(F ) The map ι 1 is induced by K → (K × K, ν) where ν(x, y) = y, γ K (x) , and the map ρ 1 carries every C 4 -Galois algebra (Q, ν) to its discriminant ∆(Q) (which is isomorphic to the quadratic subalgebra Q ν 2 , see Proposition 5.10).
Remark.The algebra K × K contains K and F × F as quadratic subalgebras.However, Galois theory shows that if (Q, ν) is a C 4 -Galois algebra and Q is a field, then Q contains a unique quadratic extension of F .
In the rest of this section, we give an explicit description of H 1 (Γ, C 4 ) and use it to parametrize C 4 -Galois algebras up to isomorphism.The description depends in an essential way on whether the characteristic is 2 or not.Remark.Another description of H 1 (Γ, µ 4[S] ) is given in [3, (30.13)].
(See [2] for a proof without cohomology and, more generally for a class of commutative rings in which 2 is invertible.)We give an explicit description of this correspondence.
Let i = √ −1 ∈ S. For λ ∈ F × and s = s 1 + is 2 ∈ S × , let d = N S/F (s) = s 2 1 + s 2 Proof.We first show that Q λ,s is a quartic étale F -algebra.Let ω, ξ, η ∈ Q λ,s be the images of W , X, Y respectively.The algebra If s 2 = 0, then d = s 2 1 , hence F [ω] F × F and we may identify ξ and η to s 1 ( √ λ, 0) and s 1 (0, √ λ) in Therefore, in each case Q λ,s is a quartic étale F -algebra, and the fact that the subalgebra fixed under ν λ,s is F is easily verified.
Proof.The first claim follows from the exact sequence of (additive) Γ-modules: where ι(s) = (0, s) and π(t, s) = t, and the fact that H 1 (Γ, F s ) = 0 (by the additive version of Hilbert's Theorem 90).The last claim follows from the exactness of (32) and from the first claim.

π←−
Z of degree 2, we may consider Y π2 * π ← −− − (X × Y ) * Y Z where Y π2 ←− X × Y is the projection map.Abusing notation, we write simply Y π ← − X * Z for this double covering.The proof of the following easy proposition is omitted: Proposition 1.5.Let X be a set of two elements with trivial Γ-action, and let Y π ← − Z be a double covering.(a)The covering Y π * π two double coverings of Y .Denote simply by the canonical automorphisms γ Z/Y and γ Z /Y and also the canonical automorphisms of ∆(Z) and ∆(Z ).For {z 1 , . . ., z n } ∈ Ω(Z/Y ) and {z 1 , . .

( a )
The F -algebra E * E is split: E * E F × F .(b) If the algebra E is split, then E * E E (not canonically).(c) The operation * defines a group structure on the set Quad(F ).
Since the Γ-action on Ω(Z/Y ) and S(Z/Y ) (if d = 2) are induced by the Γ-action on Z/Y through ω Z/Y and σ Z/Y respectively, the morphisms ω 1 Z/Y and s 1 Z/Y map the isomorphism class of the covering Z/Y to the isomorphism class of the Γ-sets Ω(Z/Y ) and S(Z/Y ), respectively.Recall also from §3.1 the canonical homomorphism β Z/Y : S Z/Y → S Y which maps every permutation of a covering to the induced permutation of the base.Let T Z/Y = ker β Z/Y .This is the group of automorphisms over Y of the covering Z/Y , hence H 1 (Γ, T Z/Y ) is in one-to-one correspondence with the set of isomorphism classes over Y of coverings of degree d of Y , where the Γ-action on Y is trivial.The case of non-trivial Γ-action can be taken into account by twisting, see [3, §28.C].If Z/Y is a covering of degree d, we define a non-trivial action of Γ on S Z/Y by conjugation: the action of Γ on Z/Y is a group homomorphism α : Γ → S Z/Y , and we define, for γ ∈ Γ and f

2 3 Γ: Set 4 Γ → Cov 2
of double coverings of Γ-sets of three elements, with morphisms of coverings.There are functors Λ ) = S(Z/Y ).Proposition 4.2 yields a natural equivalence between S • Λ and the identity on Set 4 Γ .To investigate the composition Λ • S, suppose Y π ← − Z is a double covering of a Γ-set Y with |Y | = 3. (See Example 1.3 for an explicit description of Ω(Z/Y ) and S(Z/Y ).)We may consider Z as the set of faces of a cube, Y as the set of directions of edges, and π as the map which carries each face to the orthogonal direction.Then Ω(Z/Y ) is the set of (unordered) triples of faces which are not pairwise parallel.Since the faces in each such triple meet at one vertex, we may view Ω(Z/Y ) as the set of vertices of the cube.The map γ Z/Y carries each vertex to its opposite, hence S(Z/Y ) is the set of diagonals of the cube.As in the discussion before Proposition 4.2, we may then identify Λ 2 S(Z/Y ) with the set of diagonal planes and R S(Z/Y ) with the set of edge directions.It is then clear that R S(Z/Y ) is canonically identified with Y , but there is no canonical identification of Λ 2 S(Z/Y ) with Z.

Proposition 4 . 3 .Corollary 4 . 4 .Theorem 4 . 5 .
The map Ψ defines an isomorphism of coverings betweenR S(Z/Y ) ε ← − ∆(Z) * Λ 2 S(Z/Y ) and Y π ← − Z.Proof.The map Ψ is clearly equivariant.The other properties are checked by direct inspection.This proposition shows that Λ • S is not equivalent to the identity.However, when ∆(Z) is a trivial Γ-set the proposition yields an isomorphism between Z/Y and Λ • S(Z/Y ): If the Γ-action on ∆(Z) is trivial, then there is an isomorphism of coverings betweenR S(Z/Y ) ε ← − Λ 2 S(Z/Y ) and Y π ← − Z.Proof.This readily follows from Proposition 4.3 and Proposition 1.5(b).Corollary 4.4 applies in particular to double coverings of the form Λ 2 (X)/ R(X), for X a Γ-set with |X| = 4, by Proposition 4.1.Therefore, Λ • S(X) X.The functors Λ and S define a canonical one-to-one correspondence between the set of isomorphism classes of Γ-sets of 4 elements and the set of isomorphism classes of double coverings Z/Y of Γ-sets Y of 3 elements with trivial action on ∆(Z).

Theorem 4 . 6 .
The functors Λ and Ŝ define a canonical one-to-one correspondence between the set of isomorphism classes of pairs of Γ-sets (U, X) with |U | = 2 and |X| = 4 and the set of isomorphism classes of double coverings of Γ-sets with three elements.
are the embeddings of (5).Hence we have Σ(B/A) = Σ 3 (A)[b 1 , b 1 , . . ., b 3 , b 3 ], by Proposition 3.7, and S 3 acts on Σ(B/A) through the action on Σ 3 (A) and by permuting the b i and the b j .The algebra Ω(B/A) is generated over F by all the polynomials in the b i and the b j which are symmetric under S 3 .In particular
20) above are equal, and this condition is equivalent to δ b = δ b .On the other hand, b generates B over A if and only if

F = 2 ,
we may simplify the results above by a specific choice of generating element b.Let b ∈ B be such that b = −b and assume that a = b 2 ∈ A generates A. Proposition 4.17.( char F = 2) With the notation above, )[ω b ].The formula for the minimal polynomial of δ b (resp.ω b ) follows from (21) (resp.Proposition 4.16).One can also repeat the proof of Proposition 4.16 with the special choice of b.Corollary 4.18.( char

Proof.
The first part readily follows from Proposition 4.17.If N (a) = ν 2 , then ∆(B) F ×F .Let d, d be the minimal idempotents of ∆(B).By Proposition 4.14, we have Ω(B/A) S(B/A) × S(B/A), and we may identify dΩ(B/A) and d Ω(B/A) with S(B/A).Since dδ b = ±4νd and d δ b = ∓4νd , the minimal polynomials of dω b and d λ b are X 4 − 2T (a)X 2 ± 8νX + T (a 2 ) − 2S(a).Assume now char F = 2. Let b be a generating element for B over A with b = b + 1, hence b 2 + b ∈ A. Let a = b 2 + b, i.e., using the notation ℘ for the map x → x 2 + x, a = ℘(b) ∈ A. Assume moreover a generates A, hence A = F [a] and B = F [℘ −1 (a)].

Corollary 4 .
20. ( char F = 2) With the same notation as in Proposition 4.19, the discriminant ∆(B) is split if and only if T (a) ∈ ℘(F ).If T (a) = ℘(ν) for some ν ∈ F , then S(B/A) is generated over F by an element whose minimal polynomial isX 4 + X 3 + ν 2 X 2 + ν 2 + S(a) X + ν + S(a) S(a) + N (a).Proof.The first part readily follows from Proposition 4.19.If T (a) = ℘(ν), then ∆(B) = F × F .Let d, d be the minimal idempotents of ∆(B) ⊂ Ω(B/A).We may assume d = δ b + ν and d = δ b + ν + 1, hence dδ b = (ν + 1)d.As in the proof of Corollary 4.18, we may identify dΩ(B/A) and d Ω(B/A) with S(B/A), and it follows from Proposition 4.19 that the minimal polynomial of dµ b is as stated.Combining the results of subsections 4.3 and 4.4 we get: Proposition 4.21.Let Q be a quartic étale algebra.(a) If char F = 2, let x ∈ Q be a generator such that T Q/F (x) = 0.There exists an isomorphism φ 1 :

5 . 2 . 4 : 5 . 4 .
(a) the Γ-action factors through a cyclic subgroup C 3 ; (b) the extension L/F is Galois (hence cyclic); (c) ∆(Q) F × F .Action through D 4 .Suppose now that the action of Γ factors through a Sylow 2-subgroup of S X , i.e. through a dihedral subgroup D 4 .Since the Sylow 2subgroups of S X are the isotropy groups of partitions of X into 2-element subsets, there is such a partition which is invariant under Γ.This observation characterizes the case where Γ acts through D Proposition For a set X ∈ Set 4 Γ , the following conditions are equivalent: (a) the Γ-action factors through a Sylow 2-subgroup of S X ; (b) the Γ-action leaves a point of R(X) fixed; (c) R(X) { * } ∆(X); (d) X is a double covering of a set of two elements, i.e. there exists a map (D ← X) ∈ Cov 2 2 Γ .Proof.The points of R(X) are the partitions of X into 2-element subsets, hence (a) ⇐⇒ (b).The implication (c)⇒(b) is clear, and (b)⇒(c) follows from Proposition 4.1.If D = {x 1 , x 2 }, {x 3 , x 4 } ∈ R(X) is fixed under Γ, then the canonical map D ← X which carries x 1 , x 2 to {x 1 , x 2 } and x 3 , x 4 to {x 3 , x 4 } is a double covering, hence (b)⇒(d).Finally, (d)⇒(a) follows from S 2 S 2 D 4 .
R(X) = {D} M and R( X) = {M } D, and there are natural maps D ← X and M ← X which show X and X are double coverings of D and M respectively.By Proposition 4.1, Equations (25) yield canonical isomorphisms M = ∆(X) and D = ∆( X).
(a) ⇐⇒ (b).The equivalence of (b) and (c) is clear: take for D 1 , D 2 and D 3 the fibers of the canonical map R(X) ← Λ 2 (X).The equivalence (b) ⇐⇒ (d) readily follows from Proposition 5.4.If the equivalent conditions of the proposition hold, then the set X of Proposition 5.5 can be arbitrarily chosen as D 1 D 2 , D 1 D 3 or D 2 D 3 .If we choose X = D 1 D 2 , Proposition 5.5 yields ∆( X) = D 3 .On the other hand, it is easily checked that ∆(D 1 D 2 ) D 1 * D 2 , hence D 1 * D 2 D 3 and therefore the Γ-action on D 1 * D 2 * D 3 is trivial.

ι 1 −→ 1 −
Cycl 4 (F ) ρ → Quad(F ) δ − → Br(F ), where Br(F ) is the Brauer group of F and δ maps K = F [ √ d] to the Brauer class of the quaternion algebra (−1, d) F .Of course, this result is well-known and has an easy cohomological proof.
which is a covering of degree d.Extensions of degree 2 are called quadratic extensions.Suppose B/A is an extension of degree d.Let dim F A = n (hence dim F B = nd), and let s A n ∈ (A ⊗n ) Sn be the idempotent defining Σ n (A), see §2.1.Then i ⊗n be the set of arrays (ζ ij ) 1≤i≤d 1≤j≤n of pairwise distinct elements of Z such that π(ζ ij ) depends only on j for i = 1, . . ., n.The set Σ(Z/Y ) is a S d S n -torsor.Its isomorphism class, viewed as an element of H 1 (Γ, S d S n ), corresponds to the isomorphism class of the covering Y π ← − Z.The nd projections (s 1 s 1 − s 2 s 2 )ω .Proof.The "only if" part was shown in the last lines of the proof of Proposition 6.2.(Alternately, it follows from Propositions 5.10 and 5.11.)The "if" part follows from 6.2.Characteristic 2. Cyclic Galois C p n -algebras over fields over characteristic p were constructed by Witt [14, Satz 13], using Witt vectors.The group H 1 (F, C p n ) over a field of characteristic p was computed by Serre in [9, Chap.X, §3], also in terms of Witt vectors.We recall explicitly the results of Serre and Witt for the group C 4 over a field F of characteristic 2. Let W 2 (F ) be the additive group of Witt vectors of length 2. By definition we have W 2 (F ) = {(t, s) | t, s ∈ F } with the addition (t 1 , s 1 ) + .(t 2 , s 2 ) = (t 1 + t 2 , s 1 + s 2 + t 1 t 2 ).