Editing Inverse Galois problem (section)

==Partial results==
Many cases are known. It is known that every finite group is realizable over any [[Function field of an algebraic variety|function field]] in one variable over the [[complex number]]s <math>\mathbb{C}</math>, and more generally over function fields in one variable over any [[algebraically closed field]] of [[characteristic (algebra)|characteristic]] zero. [[Igor Shafarevich]] showed that every finite [[solvable group]] is realizable over <math>\mathbb{Q}</math>.<ref>Igor R. Shafarevich, ''The imbedding problem for splitting extensions'', Dokl. Akad. Nauk SSSR '''120''' (1958), 1217-1219.</ref> It is also known that every simple [[sporadic group]], except possibly the [[Mathieu group]] {{math|''M''<sub>23</sub>}}, is realizable over <math>\mathbb{Q}</math>.<ref>p. 5 of Jensen et al., 2002</ref>

[[David Hilbert]] showed that this question is related to a [[rationality question]] for {{mvar|G}}:

:If {{mvar|K}} is any extension of <math>\mathbb{Q}</math> on which {{mvar|G}} acts as an [[automorphism group]], and the [[Invariant theory|invariant field]] {{math|''K<sup>G</sup>''}} is rational over {{nowrap|<math>\mathbb{Q}</math>,}} then {{mvar|G}} is realizable over {{nowrap|<math>\mathbb{Q}</math>.}}

Here ''rational'' means that it is a [[purely transcendental]] extension of <math>\mathbb{Q}</math>, generated by an [[algebraically independent]] set. This criterion can for example be used to show that all the [[symmetric group]]s are realizable.

Much detailed work has been carried out on the question, which is in no sense solved in general. Some of this is based on constructing {{mvar|G}} geometrically as a [[Galois covering]] of the [[projective line]]: in algebraic terms, starting with an extension of the field <math>\mathbb{Q}(t)</math> of [[rational function]]s in an indeterminate {{mvar|t}}. After that, one applies [[Hilbert's irreducibility theorem]] to specialise {{mvar|t}}, in such a way as to preserve the Galois group.

All permutation groups of degree 23 or less, except the [[Mathieu group]] {{math|''M''<sub>23</sub>}}, are known to be realizable over {{nowrap|<math>\mathbb{Q}</math>}}.<ref>{{Cite web|url=http://galoisdb.math.upb.de/|title=Home|website=galoisdb.math.upb.de}}</ref> <ref>{{Cite web|url=https://arxiv.org/abs/2411.07857|title=17T7 is a Galois group over the rationals}}</ref>

All 13 non-[[abelian group|abelian]] [[simple group]]s smaller than PSL(2,25) (order 7800) are known to be realizable over {{nowrap|<math>\mathbb{Q}</math>.}}<ref>Malle and Matzat (1999), pp. 403-424</ref>