Proposition 2.2

We have , hence is injective and is surjective.

Proof

Let K be a cubic field and an integral basis. We have (for example) . If K is a Galois cubic field, then . Otherwise, is a number field of degree 6, with Galois group isomorphic to , and is fixed by the transposition of order 2, hence belongs to a cubic subfield, and so is isomorphic to K. In fact, by choosing the numbering such that and , is even equal to K, not only conjugate to it. Note also that we cannot have otherwise would be reducible.