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If the fixed field of this action is M, then, by the fundamental theorem of Galois theory, the Galois group of F/M is G. On the other hand, it is an open problem whether every finite group is the Galois group of a field extension of the field Q of the rational numbers.
In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory .
One of the important structure theorems from Galois theory comes from the fundamental theorem of Galois theory. This states that given a finite Galois extension K / k {\displaystyle K/k} , there is a bijection between the set of subfields k ⊂ E ⊂ K {\displaystyle k\subset E\subset K} and the subgroups H ⊂ G . {\displaystyle H\subset G.}
They generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields, discovered by the French mathematician Évariste Galois. A Galois connection can also be defined on preordered sets or classes; this article presents the common case of posets. The literature contains two closely related notions of ...
In mathematics, Galois theory is a branch of abstract algebra. At the most basic level, it uses permutation groups to describe how the various roots of a given polynomial equation are related to each other.
For example, if L is a Galois extension of a number field K, the ring of integers O L of L is a Galois module over O K for the Galois group of L/K (see Hilbert–Speiser theorem). If K is a local field, the multiplicative group of its separable closure is a module for the absolute Galois group of K and its study leads to local class field theory.
Évariste Galois (/ ɡ æ l ˈ w ɑː /; [1] French: [evaʁist ɡalwa]; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a problem that had been open for 350 years.
An Extension of the Galois Theory of Grothendieck. Memoirs of the American Mathematical Society. ISBN 0-8218-2312-4. Borceux, F.; Janelidze, G. (2001). Galois theories. Cambridge University Press. ISBN 0-521-80309-8. (This book introduces the reader to the Galois theory of Grothendieck, and some generalisations, leading to Galois groupoids.)