Invariant Theory of Finite Groups
Invariant Theory of Finite Groups
Invariant Theory of Finite Groups
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<strong>Invariant</strong> <strong>Theory</strong> <strong>of</strong> <strong>Finite</strong> <strong>Groups</strong> – p. 19/2Noether’s theoremTheorem (Emmy Noether (1926)): If G is a finite group, then A G isa finitely generated F-algebra.Pro<strong>of</strong>: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].B is a noetherian F - algebra (Hilbert’s theorem)B ≤ A is finite integral extension;A = 〈 ∏ di=1 xm ii | m i < n〉 B (since f i (x i ) = 0);⇒ B A is a finitely generated, so a noetherian B - module;
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