Editing Inverse Galois problem (section)

{{Short description|Unsolved problem in mathematics}}
{{unsolved|mathematics|Is every [[finite group]] the [[Galois group]] of a [[Galois extension]] of the [[rational number]]s?}}
In [[Galois theory]], the '''inverse Galois problem''' concerns whether or not every [[finite group]] appears as the [[Galois group]] of some [[Galois extension]] of the [[rational number]]s <math>\mathbb{Q}</math>. This problem, first posed in the early 19th century,<ref>{{Cite web |title=Mathematical Sciences Research Institute Publications 45 |url=http://library.msri.org/books/Book45/files/book45.pdf |website=MSRI |access-date=2016-04-17 |archive-date=2017-08-29 |archive-url=https://web.archive.org/web/20170829003635/http://library.msri.org/books/Book45/files/book45.pdf |url-status=dead }}</ref> is unsolved.

There are some [[permutation group]]s for which [[generic polynomial]]s are known, which define all [[algebraic extension]]s of <math>\mathbb{Q}</math> having a particular [[group (mathematics)|group]] as Galois group. These groups include all of degree no greater than {{math|5}}. There also are groups known not to have generic polynomials, such as the cyclic group of [[order (group theory)|order]] {{math|8}}.

More generally, let {{mvar|G}} be a given finite group, and {{mvar|K}} a field. If there is a Galois extension field {{math|''L''/''K''}} whose Galois group is [[group isomorphism|isomorphic]] to {{mvar|G}}, one says that '''{{mvar|G}} is realizable over {{mvar|K}}'''.