section_path: Contributions to mathematics and physics/Second epoch (1920–1926)/Invariant theory of finite groups
section_name: Invariant theory of finite groups
paragraph_text: Invariant theory of finite groups
  
  Techniques such as Hilbert's original non-constructive solution to the finite basis problem could not be used to get quantitative information about the invariants of a group action, and furthermore, they did not apply to all group actions. In her 1915 paper, Noether found a solution to the finite basis problem for a finite group of transformations G acting on a finite-dimensional vector space over a field of characteristic zero. Her solution shows that the ring of invariants is generated by homogeneous invariants whose degree is less than, or equal to, the order of the finite group; this is called Noether's bound. Her paper gave two proofs of Noether's bound, both of which also work when the characteristic of the field is coprime to | G | ! {\displaystyle \left|G\right|!} (the factorial of the order | G | {\displaystyle \left|G\right|} of the group G). The degrees of generators need not satisfy Noether's bound when the characteristic of the field divides the number | G | {\displaystyle \left|G\right|}, but Noether was not able to determine whether this bound was correct when the characteristic of the field divides | G | ! {\displaystyle \left|G\right|!} but not | G | {\displaystyle \left|G\right|}. For many years, determining the truth or falsehood of this bound for this particular case was an open problem, called "Noether's gap". It was finally solved independently by Fleischmann in 2000 and Fogarty in 2001, who both showed that the bound remains true.
  
  In her 1926 paper, Noether extended Hilbert's theorem to representations of a finite group over any field; the new case that did not follow from Hilbert's work is when the characteristic of the field divides the order of the group. Noether's result was later extended by William Haboush to all reductive groups by his proof of the Mumford conjecture. In this paper Noether also introduced the Noether normalization lemma, showing that a finitely generated domain A over a field k has a set { x,..., x } of algebraically independent elements such that A is integral over k.