EXPLICIT DECOMPOSITION OF A RATIONAL PRIME IN A CUBIC FIELD

We give the explicit decomposition of the principal ideal 〈 p 〉 ( p prime) in a cubic field.


Introduction
Let K be an algebraic number field.Let O K denote the ring of integers of K. Let d(K) denote the discriminant of K. Let θ ∈ O K be such that K = Q(θ).The minimal polynomial of θ over Q is denoted by irr Q (θ).The discriminant D(θ) and the index ind(θ) of θ are related by the equation If p is a prime not dividing ind(θ), then it is well known that the following theorem of Dedekind gives explicitly the factorization of the principal ideal p of O K into prime ideals in terms of the irreducible factors of irr Q (θ) modulo p; see, for example, [3, Theorem 10.5.1, page 257].
Theorem 1.1.Let K = Q(θ) be an algebraic number field with θ ∈ O K .Let p be a rational prime.Let (1.2) , where Z p = Z/ pZ.Let where g 1 (x),...,g r (x) are distinct irreducible polynomials in Z p [x], and e 1 ,...,e r are positive integers.For i = 1,2,...,r, let f i (x) be any polynomial of Z[x] such that fi = g i and deg( f i ) = deg(g i ).Set P i = p, f i (θ) , i = 1,2,...,r. (1.4) If ind(θ) ≡ 0 (mod p), then P 1 ,...,P r are distinct prime ideals of O K with (1.5) On the other hand if p is a prime dividing ind(θ), no such general theorem is known which gives the prime ideals explicitly, and all that is available in general is the Buchmann-Lenstra algorithm [4, page 315] for decomposing a prime in a number field.If p is not a common index divisor of K, then there exist elements φ ∈ O K for which K = Q(φ), and p ind(φ), and we can apply Dedekind's theorem to obtain the prime ideal factorization of p from the minimal polynomial irr Q (φ).However given θ it is not easy to determine such an element φ in general.Moreover when p is a common index divisor of K, no such element φ exists and Dedekind's theorem cannot be applied.
In this paper we treat the case when K is a cubic field and p is a prime dividing ind(θ).When p is a common index divisor of K (the only possibility is p = 2), we quote the results in [2].When p is not a common index divisor, we construct an element φ ∈ O K such that K = Q(φ) and p ind(φ) and then apply Dedekind's theorem to obtain the prime ideal factorization of p .Our construction of φ was guided by the p-integral bases of K given by Alaca [1].We give explicitly the prime ideals in the factorization of p into prime ideals in O K .The form of the prime ideal factorization has been given by Llorente and Nart [6, Theorem 1, page 580] and we make use of their results.A method for factoring all primes in a cubic field is given in [5, pages 119-121].It is well known that K can be given in the form K = Q(θ), where θ is a root of the irreducible polynomial so that irr Q (θ) = f (x).Moreover it is further known that a and b can be chosen so that there are no primes p with p 2 | a and p 3 | b.We have Let ν p (k) denote the largest nonnegative integer m such that p m divides the nonzero integer k.From (1.1) we deduce that We set .
It is easy to show that the minimal polynomial of α over Q is and that disc q(x) = 3 6 b 2 D(θ) 3 . (1.12)

Case A1
In this case we can define integers A and B by a = 4A and b We have p(x) ≡ x 3 (mod2).As 2 2 disc p(x) , 2 2 d(K), we have 2 ind(φ), so that by Theorem 1.1, (2.3)

Cases A4, A8
In these cases 2 is a common index divisor and we can appeal to [6, Theorem 4, page 585] for the results.

Case B3
In this case we have Clearly 9 | 3a − b and 9 | 3a + b.However 27 cannot divide both of 3a − b and 3a + b as their sum 6a is not divisible by 27.Hence we can define We note that if 27 | b we can choose either value of θ 2 /3 ± θ for φ.Set subject to the remark above, so that The minimal polynomial of φ is

Case B4
In this case we have 9 | a + 1 − b 2 .We set (8.1) First we consider the case 9 a + 1 − b 2 .The minimal polynomial of φ is so that p(x) ∈ Z[x] and φ ∈ O K .We have where

Case B8
Here 3 is a prime ideal.

Case C3
Similarly to case B6 this case can be reduced to case C2.
.1, 1.2, 1.3, and no others.We abbreviate r ≡ s (modm) by r ≡ s(m).In the sixth column of each table we give the form of the prime ideal factorization from the work of Llorente and Nart [6, Theorem 1, page 580].However, Llorente and Nart did not give the prime ideals explicitly.We give explicit formulae for these prime ideals in the seventh column of each of Tables 1.1, 1.2, and 1.3.It is convenient to set