Let be a finite Galois extension. The Galois group acts on coordinatewise. Let be a closed algebraic set of . Prove that the following are equivalent:
- the set can be defined by polynomials in ;
- the set is invariant under the -action.
Proof. Suppose vanishes on . Let , and set . If is defined by polynomials in then and so is Galois-invariant.
Conversely, if is Galois-invariant, then if vanishes on then does also. Denoting to be the elementary symmetric polynomials of for , each and they vanish on . We claim that these polynomials define . For, if every vanished at a point , then every is zero for they are the roots of the polynomial . So the define .