Invariant Theory of Finite Groups

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Invariant Theory of Finite Groups – p. 19/2Noether’s theoremTheorem (Emmy Noether (1926)): If G is a finite group, then A G isa finitely generated F-algebra.Proof:Let A := F[x 1 ,...,x d ] and |G| = n.For i = 1,...,d setf i := ∏ g∈G (T − gx i) = T n + ∑ n−1l=0 b i,lT l ∈ A G [T].
Invariant Theory of Finite Groups – p. 19/2Noether’s theoremTheorem (Emmy Noether (1926)): If G is a finite group, then A G isa finitely generated F-algebra.Proof:Let A := F[x 1 ,...,x d ] and |G| = n.For i = 1,...,d setf i := ∏ g∈G (T − gx i) = T n + ∑ n−1l=0 b i,lT l ∈ A G [T].Define B := F[b i,l | i = 1,...,d, l = 0,...,n − 1].
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