On cubic Galois field extensions

Abstract

We study Morton's characterization of cubic Galois extensions F /K by a generic polynomial depending on a single parameter s ∈ K . We show how such an s can be calculated with the coefficients of an arbitrary cubic polynomial over K the roots of which generate F . For K = Q we classify the parameters s and cubic Galois polynomials over Z, respectively, according to the discriminant of the extension field, and we present a simple criterion to decide if two fields given by two s-parameters or defining polynomials are equal or not.

One Level Density for Cubic Galois Number Fields

“…Proof. As was stated above there is a one-to-one correspondence between Z/3Z subgroups of Cℓ m (Q) [3] and cubic Galois extensions of Q unramified outisde of the primes dividing m. There is a one-to-two correspondence between such subgroups and non-zero elements of…”

“…be the prime factorization of D i , i = 1, 2. Then by the proof of Lemma 2.3, we see that D 1 and D 2 correspond to the same cubic extension of Q if and only if the vectors (e p,1 ) and (e p,2 ) generate the same subgroup in Cℓ m (Q) [3] where m is any positive integer such that supp( suffices.…”

“…where δ m = 1 if 9|m and 0 otherwise. Finally, since Cℓ m (K) [3] is a finite abelian group, subgroups of index 3 are in one-to-one correspondence with subgroups isomorphic to Z/3Z.…”

“…Let e p be the coordinates of a element in Cℓ m (Q) [3]. Now we construct the cube-free 3-split integer as…”

“…If we let f D be the conductor of K D then we have ∆ D = f 2 D . Theorem 10 of [3] states that v p (f ) = 1 or 0 if p = 3 while v 3 (f ) = 2 or 0. Thus we get that…”