UNIFORM DISTRIBUTION OF HASSE INVARIANTS

I. Schur's study of simple algebras around the turn of the century, nd subsequent investigations by R. Brauer, E. Witt and others, were later reformulated in terms of what is now called the Schur subgroup of the Brauer group. During the last twenty years this group has generated substantial interest and numerous palatable results have ensued. Among these is the discovery that elements of the Schur group satisfy uniform distribution of Hasse invariants. It is the purpose of this paper to continue an investigation of the latter concept and to highlight certain applica- tions of these results, not only to the Schur group, but also to embeddings of simple algebras and extensions of automorphisms, among others.

R. A. MOLLIN Mollin [I0] we proved a conjecture which stated that if the center of a finite dimen- sional division algebra, D, contains no nontrivial odd order roots of unity then all finite odd order subgroups of the multiplicative group of D are cyclic.This problem is related to a conjecture of I.N.Heistein and results of S. Amitsur.Moreover in Mollin [ii] we provided sufficient conditions for the existence of a splitting field L of an absolutely irreducible character of a finite group of exponent n, such that e Q(n and IL:Q(X) mQ(X),the Schur index of X over Q.
This result is a natural outcropping of R. Brauer's well-known theorem stating that Q( splits X. Furthermore in Mollin [12 14] we used the aforementioned techniques n to answer specific questions pertaining to the structures of division algebras, the Schur index, and class field theory proper.
Given the above, it is natural to seek a more general context for uniform distri- bution.
In Mollin [15] we observed that UF(K), for K/F a finite Galois extension of number fields, could be defined as those elements of B(K) satisfying (I.D and .2) where Q is replaced by F.Moreover, as with the abelian case, we have that UF(K) is a subgroup of B(K) and S(K) is again a subgroup of UF(K).Furthermore another property which was proved in Bernard [16] to hold for S(K) carries over to UF(K).We isolate this property since it will be of independent interest later: 0.3) Let K/F be normal and suppose [A] UF(K).If and are K-primes above an F-prime P then A (R) K K and A k K have the same index, where K (respectively K) denotes the completion of K at (respectively ).The common value of indices A (R) K K for all K-primes above P is called the P-local index of A, denoted indp(A).
Now we ask whether UF(K) may somehow be salvaged when K/F is any (not necessarily normal) extension of number fields.We define UF(K) to consist of those [A] B(K) such that [A (R)  K L] UF(L), where L is the normal closure of K over F. We call UF(K) the group of algebras with uniformly distributed invariants over K relative to F. It is straightforward to check that F(K) is a subgroup of B(K) and that S(K) is in turn a subgroup of UF(K).The latter fact follows from the fact that S(K) K L is in UF(L).
However when K # L, UF(K) differs markedly from UF(L).In particular the following example shows that (1.3) fails to hold for UF(K).
EXAMPLE 1.4.Let K Q(O, 4) where is a real root of f(x) x 2 and let F Q. Then the normal closure of K over F is L Q(0,12).It can be verified that the prime 29 splits into four unramified K-primes P1,P2,P3 and P4 with Pi for i 1,2 having inertial degrees equal to one over Q, whereas P. for i 3,4 have inertial degrees equal to two over Q.
where i 1,2,3,4.By the Hasse sum theorem (see Reiner [17]) we are guaranteed that [A] B(K).Now let .be an L-prime above P. for i 1,2,3,4.It can be verified that o for i 1 2 have inertial degrees equal to 2 in L over K and for i 3,4 have inertial degrees equal to i in L over K. Hence we obtain that inv (A (R) K L) 1/2 for i 1,2,3,4.Therefore by (i.i)-.2) we have that [A K L] UQ(L), and so [A] UQ(K).However the index of A K Kpp is four, whereas the index of A (R) K Kp is two, contradicting (1.3).This completes the example.
The first main result of this paper is to provide a generalization of a theorem of E. Witt using UF(K) as a tool.From this result we will see that (1.1)-(1.2)fail to hold for IJF(K) when K L. Furthermore we use this generalized Witt theorem to develop further properties of IJF(K ).Moreover we obtain necessary and sufficient conditions, in terms of the arithmetic of the fields under consideration, for an element to exist in UF(K) with a given index.Several related properties are also developed.
Finally we generalize the concept of K-adequacy introduced in Fein et al [18] and link it to IJF(K via the arithmetic of the underlying fields in a sequence of results.

BASIC REFERENCES
For basic properties of number fields used in this paper we refer the reader to Marcus [19].For fundamental results concerning the Schur subgroup of the Brauer group the reader should consult Yamada [20].For information pertaining to properties of algebras used herein, and especially the classification of the Brauer group of a number field via Hasse invariants see the beautifully written Reiner [17].Any concepts not described in greater detail in this paper may be found in earlier work IJF(K where K/F is an extension of number fields. Let P be a K-prime with invp(A) O, and let be an L-prime above P where L is the normal closure of K/F.Set g g.c.d.(m, IL:KpI), where m is the index of A Kp. Then #m/g is in K and P N F is completely split in F(m/g).
PROOF.Let o G(L/K) and let be an L-prime with K P. Therefore inv(A K L) bo invo(A (R)K L) where Om bO'm Hence: IL:KpI invp(A) bolLo:Kplinvp (A) (nod i).Thus bo 1 (nod m/g), and so o fixes m/g for all o G(L/K)" i.e. m/g is in K.

Q.E.D.
For completeness sake we state Witt's results [21, Satz i0, Satz ii, p.243] in succinct form as a corollary which is immediate from the theorem.COROLLARY 3 3 Let m/g pa where p is a prime and suppose n is the largest p p-power root of unity in K If P is completely split in F( c) but not in F( c+l) p P then a <_ min{n,c}.
The next result which will prove to be useful later in the paper generalizes IL:Ko I, where (respectively ) is any L-prime above P(respectively ).If  IL:Kplinvp(A) bolL:Klinvi(A (rood i).
Since [L'Kp[ ]L:K[ then we have that invp(A) inv(A) if and only if bo E 1 (mod m/g) which in turn holds if and only if o G(L/F(m/g)).However by theorem 3.1, P fl F is completely split in F(m/g).Hence o G(L/F(m/g)) if and only If p F(m / po F( ).
Q.E.D. g /g W note that Theorem 3. siows that (i.I) does not hod for UF(K when K/F is non- normal.Moreover kl.does not generalize to UF(K) where oof (I.i)is interpreted as an embedding of K into the complex field C.This may be illustrated by considering example (1.4) with o C(I/K) and oy .However invp2(A) 3/4 invp3(A) 1/2 (rood i).
Now, Theorem 3.1 is the "best possible" generalization of Witt's results in the sense that we cannot hope for P Q F to be completely split in F( for m n > m/g n in general.The following example depicts this fact.EIPLE 3.5.Let be a real root of f(x) x 16   2, and let F Q().If K Q(I,# 8) then L Q(.16) is the normal closure of K over F. Let P be an F-prime above 5.Then it can be verified that P splits into two unramified K-primes 1 and (A) (A) 1/8, and inv,2_ 2 each having inertial degree two over P Define: invl -1/8 while inv,(A) 0 for all K-primes Q # i,2.Then by the Hasse sum Theorem [A] B(K).Now since there exists exactly one L-prime i" above i for i 1,2, each with inertial degree equal to 2 then invl(A K L) 1/4 and inv2(A K L) -1/4.

Thus by construction [A]
UF(K ).However, although the index of A (R) K K is 8, P is not completely split in F(8) K.This completes the example.
It is natural to ask whether the converse of Theorem 3.1 holds since we would then have a criterion, in terms of the arithmetic of K and F, for the existence of an element in IJF(K).Unfortunately the converse fails to hold as the following counter- example illustrates.EXAMPLE 3.6.Let K Q(01), F Q, and L Q(02,63) where 01 is a real root of f(x) x 2 and 02 is a real root of g(x) x ii.Then 2 splits into two K-primes PI and P2 with inertial degrees one and two respectively over F. Hence P1 splits into two L-primes ! and 2 each with inertial degree 2 over K" and P2 has one L-prime 3 above it with inertial degree one over K. Now, if there exists [A] IJF(K with invP2(A) 1/2 then [A K L] UF(L).Therefore: inv3(A L) IL3: Kp21invp2(A invp2(A) ---1/2 (rood i).Therefore ind2(A K L) 2 is forced.Thus: inv (A k L) 1/2 for i 1,2,3.However by Mollin [15, Lemma 2.8, p.259] we must i have [ inv (A (R)K L) 0 (mod I), a contradiction which completes the example.i=l Pi Now we demonstrate that under a suitable restriction we do get sufficient condi- tions in terms of the arithmetic of F.K and L to guarantee the existence of an element in F (K). THEOREM 3.7.Let L be the normal closure of K/F, an extension of number fields.

Suppose that
is the largest root of unity in F with n 2 (mod 4) If we have: n UF(K) with the index of A K K equal to r.
PROOF.By [15, Theorem 2.7, p.256] we have the existence of an element [B] IJF(I.with indp(B) r.Now, let I(B) {F-primes P: indp(B) i}.Suppose that inv,(B) a(_.)/rwhere is an L-prime above P I(B).By Mollin [15, Corollary 2.5, p.254] we have that, for any L-prime such that N K K then, inv,(B) a()/r.Therefore we set a() a(._) for all such L-primes , .above a K-prime which lies above a given P I(B).Now we define a K-central simple algebra A as follows.
For each P I(B), let inv,(A) a(_).()/r, for all K-primes above P, and let inv,(A) 0 for all K-primes not above primes in I(B), where () IL:Ki/IL:K!.
By (3) of the hypothesis we have that the index of A K K is equal to r. Moreover: 0 I inv,(B) I a()/r @ a(..)g(_)/r y invS (A) (rood i).
The final result of this section is an interesting result whidh generalizes Mollin [15, Lemma 3.1, p.262] and simplifies the proof thereof.Moreover we note that it is possible to use the following result to generalize Mollin [15, Theorem 2.10, p.260].
Q.E.D. 4. EMBEDDINGS IN SIMPLE ALGEBRAS AND EXTENSIONS OF AUTOMORPHISMS.
Let K/F be an extension of number fields, and let n be a fixed positive integer.
If D is a division algebra with [D] B(K) then we say that D is (n,F)-adequate if there exists an F-division algebra B with F.I c D c M (B) where I is the identity Since L/F i: normal then it follows rom Mollin [4, (2.2)-(2.3),p.276] that: inv(D L) inv(D . .L) inv(D K L) inv(D K L).Since g.c.d.(IL:KI,m) 1 then inv(D @K L) # inv(D L) implies g.c.d. (ID:KI,ID I:K I) @ 1 which contradicts that D D is a division algebra, (see (Reiner [17].)We note that the above proof essentially uses the idea of Fein [22].
Q.E.D. Now we use Lemma 4.1 to obtain a generalization of Mollin [15, Theorem 3.2, p.263].THEOREM 4.2.Suppose K./F for i 1 2 are extensions of number fields with N being the normal closure of KI/F and assume that KIK2/F is normal.Suppose D is a division algebra with [D]E UF(KI) and which has index m.Suppose D is (n,K2)- adequate for a given positive integer n such that g.c.d.(m, INIK2:KIK21) i.If g is in K g.c d (INI:KII,m) then m/g KIK2] UF(KIK2) and so we have PROOF By Theorem 3 8 we have that [D KI UF(KIK2) c_ UK2(KIK2).Now let P and be any two KiK2-primes with P K 2 K 2.
KIK 2) inv(D (R)KI KIK2).Therefore Now we invoke Lemma 4 1 to get that invp(D (R)KI since KIK2/F is normal then we may invoke Corollary 3.4 to get P K2(m/g) 0 K 2(m/g).However P and were arbitrarily chosen subject only to P K2" N K.
is in K 2. Q.E.D.By Theorem 3 i, P K 2 is completely split in K2(#m/g ).Hence m/g We note that immediate consequences of Theorem 4.2 are Mollin [8, Theorem 3.1, p.175], and Fein et al [24, Theorem I, p.305].Moreover the above proof is shorter and more straightforward than the latter two cases.
Finally we present the following generalization of Mollin [i0, Theorem 4.5, p.245].The result is virtually immediate but we present it since it may be of independent interest.Aut(K) (reap.Aut(A)) refers to the automorphism group of K(resp.A).THEOREM 4.3.Let K/F be an extension of number fields and assume that the fixed field of Aut(K) is contained in F. If [A] E UF(K) with index n then o Aut(K) extends to Aut(A) if and only if o fixes n PROOF.Since UF,(K) _c UF(K) where F' is the fixed field of Aut(K) then the result follows immediately from Mollin [i0, Theorem 4.5, p.245].
Q.E.D. We conclude with a note that it is an open question as to whether other results may be generalized to UF(K) when K/F is not normal.Some such examples are Mollin [6,    .It should also be noted that recently Greenfield [25] has done some work on uniform distribution from a different perspective mode;led after Mollin [15].

COROLLARY 3 . 2 .
If K/Q is finite abelian, [A] S(K) and p is an odd prime with ind (A) m then p i (mod m).If p 2 then ind (A) i or 2. P P We maintain the notation of the Theorem 3.1 in the following results.The first result, which is immediate, generalizes Mollin [15,Corollary 2.4, p.254].
i30 R.A. MOLLIN PROOF.Suppose o G(L/F) such that .We have inv(A ' L) b inv(A L) (rood i) where o b Ths we have by Mollin [4 (2 3) p.276] m/g m/g that: (i)   is in K where n divides m and; ^m (2) P is a K-prime such that N F P is completely split in F( ), and m(3) g.c.d.(r, {L:K I) I, [A] F(KI) and g.c.d. (m, ININ2:N21) i then [A KI KIK 2] F(KIK2 )- .d. (m, ININ2:KII) and let o G(N2/F) with m/g_ m/g Denote an extension of o to G(NIN2/F) by .Since [A (R)KI N I] UF(NI) implies that n matrix of M (B) the full ring of n n matrices with entries from B. This concept n generalizes that of Fein et al [18] which is the n R. A. MOLLIN result is a generalization of Fein [22, Proposition 3, p.438].LEMMA 4.1.Let K/F be an extension of number fields with L being the normal closure of K/F.Suppose [D] B(K) where D is a division ring of index m such that g.c.d.(m, IL:KI) I.If D is (n,F)-adequate for a given positive integer n then inv(D K L) inv(D L) for all L-primes and such that F N F. PROOF.Let A Mn(B) where D is embedded in A. If C CA(K) denotes the centra- lizer of K in A then by Reiner [17, Corollary 7.14, p.96] we have [C] [B (R)F K]- Moreover by Albert [23, Theorem 13, p.53] we have [C] [D K DI] where D is a division ring with [DI] Theorem 4.5, p.476], [7, Theorem 2.1, p.202], [2, Theorem 1.2, p.262], and [12, Theorem i, p.i075]