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Questions tagged [galois-theory]

The study of connections between field theory and group theory through the idea of Galois groups and further concepts.

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How did the modern understanding of Galois theory come about?

The "modern" understanding of the Galois group of a polynomial is as automorphisms of the splitting field of the polynomial which keep the base field fixed. These concepts were unknown to Galois, who ...
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1answer
102 views

Did Galois make use of the concept of a basis?

I've been reading Galois' First Memoir, where he introduces Galois Theory by giving a sufficient and necessary condition for a polynomial to be solvable by radicals. The proofs are a bit sketchy and ...
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1answer
445 views

History of Galois Theory after Galois

Galois theory occupies a rather central place in modern number theory, from class field theory, to the Langlands program, to the ideas found in Grothendieck's Esquisse d'un Programme. But the ...
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2answers
750 views

Did Evariste Galois create the entire group structure concept?

Did Evariste Galois create the entire group structure concept? If yes, were "super-sets" of groups (e.g. rings or vector spaces) created on top of Galois's work? When and by who? If no, did Galois ...
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0answers
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Who is the first to give the proof of insolvability of quintic functions using Galois theory?

The first correct proof of the insolvability of the quintic is due to Abel. But my question is who gave the proof of insolvability of the quintic using Galois theory? Does Abel know about Galois ...
12
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1answer
371 views

What does Lagrange mean in this passage from Reflexions sur la résolution?

I'm having difficulty with section 6 of Lagrange's Réfléxions sur la résolution algébrique des équations. That's page 11 of the paper, 215 of his Oeuvres. Annoyingly, this is one of the most critical ...
13
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1answer
175 views

When was the modern field theory approach to Galois theory developed?

It is well known that Galois, and other mathematicians around that time, considered Galois groups to be permutation groups and approached Galois theory in this manner. At some point the theory took a ...