Editing Emmy Noether (section)

====Algebraic invariant theory====
[[File:Emmy Noether - Table of invariants 2.jpg|thumb|250px|right|Table&nbsp;2 from Noether's dissertation{{Sfn|Noether|1908}} on invariant theory. This table collects 202 of the 331&nbsp;invariants of ternary biquadratic forms. These forms are graded in two variables ''x'' and ''u''. The horizontal direction of the table lists the invariants with increasing grades in ''x'', while the vertical direction lists them with increasing grades in ''u''.]]
Much of Noether's work in the first epoch of her career was associated with [[invariant theory]], principally [[algebraic invariant theory]]. Invariant theory is concerned with expressions that remain constant (invariant) under a [[group (mathematics)|group]] of transformations.{{sfn|Dieudonné|Carrell|1970}} As an everyday example, if a rigid [[metre-stick]] is rotated, the coordinates of its endpoints change, but its length remains the same.  A more sophisticated example of an ''invariant'' is the [[discriminant]] {{math|''B''<sup>2</sup> − 4''AC''}} of a homogeneous quadratic polynomial {{math|''Ax''<sup>2</sup> + ''Bxy'' + ''Cy''<sup>2</sup>}}, where {{mvar|x}} and {{mvar|y}} are [[indeterminate (variable)|indeterminate]]s. The discriminant is called "invariant" because it is not changed by linear substitutions {{math|''x'' → ''ax'' + ''by''}} and {{math|''y'' → ''cx'' + ''dy''}} with determinant {{math|1=''ad'' − ''bc'' = 1}}. These substitutions form the [[special linear group]] {{math|''SL''<sub>2</sub>}}.<ref>{{cite web|last1=Lehrer|first1=Gus|title=The fundamental theorems of invariant theory classical, quantum and super|url=https://www.math.auckland.ac.nz/~dleemans/NZMRI/lehrer.pdf|publisher=[[University of Sydney]]|access-date=9 February 2025|archive-url=https://archive.today/20250209193607/https://www.math.auckland.ac.nz/~dleemans/NZMRI/lehrer.pdf|archive-date=9 February 2025|page=8|date=January 2015|url-status=live|type=Lecture notes}}</ref>

One can ask for all polynomials in {{mvar|A}}, {{mvar|B}}, and {{mvar|C}} that are unchanged by the action of {{math|''SL''<sub>2</sub>}}; these turn out to be the polynomials in the discriminant.{{Sfn|Schur|1968|p=45}} More generally, one can ask for the invariants of [[homogeneous polynomial]]s {{math|''A''<sub>0</sub>''x''<sup>''r''</sup>''y''<sup>0</sup> + ... + ''A<sub>r</sub>x''<sup>0</sup>''y''<sup>''r''</sup>}} of higher degree, which will be certain polynomials in the coefficients {{math|''A''<sub>0</sub>, ..., ''A<sub>r</sub>''}}, and more generally still, one can ask the similar question for homogeneous polynomials in more than two variables.{{Sfn|Schur|1968}}

One of the main goals of invariant theory was to solve the "''finite basis problem''". The sum or product of any two invariants is invariant, and the finite basis problem asked whether it was possible to get all the invariants by starting with a finite list of invariants, called ''generators'', and then, adding or multiplying the generators together.{{Sfn|Reid|1996|p=30}} For example, the discriminant gives a finite basis (with one element) for the invariants of a quadratic polynomial.{{Sfn|Schur|1968|p=45}}

Noether's advisor, Paul Gordan, was known as the "king of invariant theory", and his chief contribution to mathematics was his 1870 solution of the finite basis problem for invariants of homogeneous polynomials in two variables.{{sfn |Noether|1914|p=11}}{{Sfn |Gordan| 1870}} He proved this by giving a constructive method for finding all of the invariants and their generators, but was not able to carry out this constructive approach for invariants in three or more variables. In 1890, David Hilbert proved a similar statement for the invariants of homogeneous polynomials in any number of variables.{{Sfn|Weyl|1944|pp=618–621}}{{Sfn|Hilbert|1890|p=531}} Furthermore, his method worked, not only for the special linear group, but also for some of its subgroups such as the [[special orthogonal group]].{{Sfn |Hilbert | 1890 | p = 532}}

Noether followed Gordan's lead, writing her doctoral dissertation and several other publications on invariant theory. She extended Gordan's results and also built upon Hilbert's research. Later, she would disparage this work, finding it of little interest and admitting to forgetting the details of it.{{sfn|Dick|1981|pp=16–18,155–156}} Hermann Weyl wrote, <blockquote>[A] greater contrast is hardly imaginable than between her first paper, the dissertation, and her works of maturity; for the former is an extreme example of formal computations and the latter constitute an extreme and grandiose example of conceptual axiomatic thinking in mathematics.{{sfn|Dick|1981|p=120}}</blockquote>