# Ring is integral extension of fixed-point subring under finite automorphism group

## Statement

Let be a commutative unital ring and be a finite group acting as automorphisms on (in other words, is a finite subgroup of the automorphism group of ). Let be the subring of comprising those elements fixed by *every* element of . Then, is an integral extension of . In fact, every element of satisfies a monic polynomial over of degree equal to the order of .

## Related facts

## Proof

Let . Consider the elements for . Then, all the elementary symmetric polynomials in these elements take values in . Hence, we can construct a monic polynomial of degree equal to the order of , with all coefficients in , and whose roots are precisely the elements .