= Hilbert finite generation for reductive invariants
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Choose finitely many homogeneous positive-degree invariants generating the ideal generated by all positive-degree invariants in the ambient polynomial ring. Such a finite set exists because a polynomial ring is Noetherian. Apply the <Reynolds operator on a polynomial invariant ring> to an expression for any homogeneous invariant in that ideal. Its coefficients become invariants of smaller degree, so induction proves that the chosen set generates the entire invariant ring as an algebra.
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