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In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory , allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand.
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.)
Galois Theory (1984) [19] Galois theory is the study of the solutions of polynomial equations using abstract symmetry groups. This book puts the origins of the theory into their proper historical perspective, and carefully explains the mathematics in Évariste Galois' original manuscript (reproduced in translation). [20] [21]
In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories.
It was proved by Évariste Galois in his development of Galois theory. In its most basic form, the theorem asserts that given a field extension E / F that is finite and Galois , there is a one-to-one correspondence between its intermediate fields and subgroups of its Galois group .
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.
In his 2001 book, Galois' Theory of Algebraic Equations, he explored the evolution of algebra from ancient Babylon to the eras of Galois and Grothendieck. A review by the Mathematical Association of America said, "Anybody with an interest in algebra or the history of mathematics should look at this book.
Before Galois, there was no clear distinction between the "theory of equations" and "algebra". Since then algebra has been dramatically enlarged to include many new subareas, and the theory of algebraic equations receives much less attention. Thus, the term "theory of equations" is mainly used in the context of the history of mathematics, to ...