# Difference between revisions of "Ring is integral extension of fixed-point subring under finite automorphism group"

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

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+ | ==Related facts== | ||

+ | * [[Automorphism group acts transitively on fibers of spectrum over fixed-point subring]] | ||

==Proof== | ==Proof== | ||

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

## Latest revision as of 16:34, 12 May 2008

## 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 .