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

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Statement

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

Proof

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