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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Is there a translation of Kronecker's original work on adjoining a root of a polynomial to a field?
I would be interested in reading how Kronecker formally approached this construction, using the mathematical ideas of his time, and possibly some insight as to what he considered its philosophical ...
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Why is it called a group action?
A group action has two laws which roughly correspond to associativity and identity
$
\phi : (G : \textrm{Group}) \times (S : \textrm{Set}) \rightarrow S \\
\forall a, b : G . \forall c : S. \phi(a,\...
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Lost memoir of Évariste Galois
According to the Wikipedia article on Évariste Galois
He submitted his memoir on equation theory several times, but it was
never published in his lifetime due to various events. Though his
first ...
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Question about Hermite's 1858 solution to the quintic equation using elliptic modular functions and it's relation to Gauss' and Jacobi's work
The general quintic equation cannot be solved by radicals and is shown in a landmark and far reaching work of Galois from 1832, which became a template of modern Group theory and Galois theory. ...
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Why did Jordan introduce his canonical form?
Camille Jordan's famous canonical form for matrices over algebraically closed fields, is considered an important result nowadays, commonly taught to all students of mathematics in undergraduate linear ...
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Where are Galois's notes?
There are a number of photos of bits of Galois's notes floating around online, such as here and here. But where are they, physically? I'm hoping what remained of them wasn't tossed out once photos ...
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Did Descartes leave solving the quintic as an exercise to his readers?
In this document by Jim Brown it is claimed (on Section 3, pg 5) that:
[Descartes] believed that all polynomials of degree $>4$ could be solved with the same methods as had been applied to the ...
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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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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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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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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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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 ...
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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 ...
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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 ...
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