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

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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 $G$.
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 $g \cdot x$.