### 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.