Editing Emmy Noether (section)

===First epoch (1908–1919)===
====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>

====Galois theory====
[[Galois theory]] concerns transformations of [[field (mathematics)|number fields]] that [[permutation|permute]] the roots of an equation.<ref>{{harvnb|Stewart|2015|pp=108–111}}</ref> Consider a polynomial equation of a variable {{math|''x''}} of [[Degree of a polynomial|degree]] {{math|''n''}}, in which the coefficients are drawn from some [[ground field]], which might be, for example, the field of [[real number]]s, [[rational number]]s, or the [[integer]]s [[modular arithmetic|modulo]]&nbsp;7. There may or may not be choices of {{math|''x''}}, which make this polynomial evaluate to zero. Such choices, if they exist, are called [[root of a function|roots]].{{sfn|Stewart|2015|pp=22-23}} For example, if the polynomial is {{math|''x''<sup>2</sup> + 1}} and the field is the real numbers, then the polynomial has no roots, because any choice of {{math|''x''}} makes the polynomial greater than or equal to one.{{sfn|Stewart|2015|pp=23, 39}} If the field is [[field extension|extended]], however, then the polynomial may gain roots,{{sfn|Stewart|2015|pp=39, 129}} and if it is extended enough, then it always has a number of roots equal to its degree.{{sfn|Stewart|2015|pp=44, 129, 148}}

Continuing the previous example, if the field is enlarged to the complex numbers, then the polynomial gains two roots, {{math|+''i''}} and {{math|−''i''}}, where {{math|''i''}} is the [[imaginary unit]], that is, {{math|1=''i''<sup> 2</sup> = −1.}} More generally, the extension field in which a polynomial can be factored into its roots is known as the [[splitting field]] of the polynomial.<ref>{{harvnb|Stewart|2015|pp=129–130}}</ref>

The [[Galois group]] of a polynomial is the set of all transformations of the splitting field which preserve the ground field and the roots of the polynomial.<ref>{{harvnb|Stewart|2015|pp=112–114}}</ref> (These transformations are called [[automorphism]]s.) The Galois group of {{nowrap|{{math|''x''<sup>2</sup> + 1}}}} consists of two elements: The identity transformation, which sends every complex number to itself, and [[complex conjugation]], which sends {{math|+''i''}} to {{math|−''i''}}. Since the Galois group does not change the ground field, it leaves the coefficients of the polynomial unchanged, so it must leave the set of all roots unchanged. Each root can move to another root, however, so transformation determines a [[permutation]] of the {{math|''n''}} roots among themselves. The significance of the Galois group derives from the [[fundamental theorem of Galois theory]], which proves that the fields lying between the ground field and the splitting field are in one-to-one correspondence with the [[subgroup]]s of the Galois group.<ref>{{harvnb|Stewart|2015|pp=114–116, 151–153}}</ref>

In 1918, Noether published a paper on the [[inverse Galois problem]].<ref>{{harvnb|Noether|1918}}.</ref> Instead of determining the Galois group of transformations of a given field and its extension, Noether asked whether, given a field and a group, it always is possible to find an extension of the field that has the given group as its Galois group. She reduced this to "[[Noether's problem]]", which asks whether the fixed field of a subgroup ''G'' of the [[symmetric group|permutation group]] {{math|''S''<sub>''n''</sub>}} acting on the field  {{math|''k''(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}} always is a pure [[transcendental extension]] of the field {{math|''k''}}. (She first mentioned this problem in a 1913 paper,<ref>{{harvnb|Noether|1913}}.</ref> where she attributed the problem to her colleague [[Ernst Sigismund Fischer|Fischer]].) She showed this was true for  {{math|''n''}} = 2, 3, or 4. In 1969, [[Richard Swan]] found a counter-example to Noether's problem, with  {{math|''n''}} = 47 and {{math|''G''}} a [[cyclic group]] of order&nbsp;47<ref>{{harvnb|Swan|1969|p=148}}.</ref> (although this group can be realized as a [[Galois group]] over the rationals in other ways). The inverse Galois problem remains unsolved.<ref>{{Harvnb|Malle|Matzat|1999}}.</ref>

====Physics====
{{main|Noether's theorem|Conservation law (physics)|Constant of motion}}

Noether was brought to [[Göttingen]] in 1915 by David Hilbert and Felix Klein, who wanted her expertise in invariant theory to help them in understanding [[general relativity]],{{sfn|Gowers et al.|2008|p=800}} a geometrical theory of [[gravitation]] developed mainly by [[Albert Einstein]]. Hilbert had observed that the [[conservation of energy]] seemed to be violated in general relativity, because gravitational energy could itself gravitate. Noether provided the resolution of this paradox, and a fundamental tool of modern [[theoretical physics]], in a 1918 paper.<ref>{{harvnb|Noether|1918b}}</ref> This paper presented two theorems, of which the first is known as [[Noether's theorem]].<ref>{{harvnb|Kosmann-Schwarzbach|2011|p=25}}</ref> Together, these theorems not only solve the problem for general relativity, but also determine the conserved quantities for ''every'' system of physical laws that possesses some continuous symmetry.<ref>{{cite web |last1=Lynch |first1=Peter |author-link=Peter Lynch (meteorologist) |date=18 June 2015 |title=Emmy Noether's beautiful theorem |url=https://thatsmaths.com/2015/06/18/emmy-noethers-beautiful-theorem/ |access-date=28 August 2020 |website=ThatsMaths |archive-url=https://web.archive.org/web/20231209003118/https://thatsmaths.com/2015/06/18/emmy-noethers-beautiful-theorem/ |archive-date=9 December 2023 |url-status=live}}</ref> Upon receiving her work, Einstein wrote to Hilbert:{{blockquote|Yesterday I received from Miss Noether a very interesting paper on invariants. I'm impressed that such things can be understood in such a general way. The old guard at Göttingen should take some lessons from Miss Noether! She seems to know her stuff.<ref>{{Harvnb|Kimberling|1981|p=13}}</ref>}}

For illustration, if a physical system behaves the same, regardless of how it is oriented in space, the physical laws that govern it are rotationally symmetric; from this symmetry, Noether's theorem shows the [[angular momentum]] of the system must be conserved.{{sfn|Zee|2016|p=180}}<ref name="ledhill">{{Harvnb|Lederman|Hill|2004|pp=97–116}}.</ref> The physical system itself need not be symmetric; a jagged asteroid tumbling in space [[Conservation of angular momentum|conserves angular momentum]] despite its asymmetry. Rather, the symmetry of the ''physical laws'' governing the system is responsible for the conservation law. As another example, if a physical experiment works the same way at any place and at any time, then its laws are symmetric under continuous translations in space and time; by Noether's theorem, these symmetries account for the [[Conservation law (physics)|conservation laws]] of [[momentum|linear momentum]] and [[energy]] within this system, respectively.{{sfn|Taylor|2005|pp=268–272}}<ref>{{cite book |last=Baez |first=John C. |author-link=John C. Baez |chapter=Getting to the Bottom of Noether's Theorem |pages=66–99 |title=The Philosophy and Physics of Noether's Theorems |editor-first1=James |editor-last1=Read |editor-first2=Nicholas J. |editor-last2=Teh |year=2022 |publisher=Cambridge University Press |isbn=9781108786812 |arxiv=2006.14741}}</ref>

At the time, physicists were not familiar with [[Sophus Lie]]'s theory of [[Lie group|continuous groups]], on which Noether had built. Many physicists first learned of Noether's theorem from an article by [[Edward Lee Hill]] that presented only a special case of it. Consequently, the full scope of her result was not immediately appreciated.<ref>{{harvnb|Kosmann-Schwarzbach|2011|pp=26, 101–102}}</ref> During the latter half of the 20th century, however, Noether's theorem became a fundamental tool of modern [[theoretical physics]], both because of the insight it gives into conservation laws, and also, as a practical calculation tool. Her theorem allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it facilitates the description of a physical system based on classes of hypothetical physical laws. For illustration, suppose that a new physical phenomenon is discovered. Noether's theorem provides a test for theoretical models of the phenomenon: If the theory has a continuous symmetry, then Noether's theorem guarantees that the theory has a conserved quantity, and for the theory to be correct, this conservation must be observable in experiments.<ref name="neeman_1999" />