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Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is, in some sense, simpler and better understood.

8 votes

Grothendieck's Galois theory without finiteness hypotheses

Besides the references already given above, one can also mention §7 of SGA III, Exp.X (Caracterisation Et Classification Des Groupes De Type Multiplicatif, doi:10.1007/BFb0059008), where Grothendieck …
David Roberts's user avatar
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15 votes
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Anabelian geometry study materials?

There is this very beautiful survey Nakamura, Hiroaki; Tamagawa, Akio; Mochizuki, Shinichi The Grothendieck conjecture on the fundamental groups of algebraic curves http://www.math.sci.osaka-u.ac. …
Will Sawin's user avatar
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2 votes
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Image of spliting of short exact sequence of algebraic fundamental groups

First a remark : you have to suppose $X$ geometrically connected to get this exact sequence. I don't know what kind of understanding you expect, but the answer to the question will it always be the fu …
Niels's user avatar
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