= Hilbert finite generation for reductive invariants
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For a finite-dimensional <rational representation> of a complex <reductive algebraic group>, 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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