Editing Emmy Noether (section)

====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>