16
votes
1answer
1k views

On a theorem of Galois

I am currently teaching Galois theory and this week, I mentioned the following theorem of Galois : Let $P(x) \in \mathbf{Q}[x]$ be an irreducible polynomial of prime degree. Then $P$ is solvable by ...
5
votes
0answers
269 views

Cutting and pasting in Galois theory

I want to ask who was the first to use cut-paste construction in Galois theory. This question is motivated from the trend in contemporary Galois theory to use patching methods to construct Galois ...
19
votes
0answers
1k views

more on “Transalgebraic Theories” (a 19th century yoga)?

Among the talks at occasion of the Galois Bicentennial, one is about "Transalgebraic Theories". Unfortunately I found only this article describing that fascinating idea as " an extremely powerful ...
5
votes
1answer
1k views

Motivation for the proof of Hilbert's Theorem 90

The proof of Hilbert's Theorem 90 about cyclic extensions goes like this: Let $\sigma$ be the generator of the Galois group of order $n$ and let $b$ have norm $1$, i.e. $b \sigma(b) \cdots ...
11
votes
1answer
1k views

History of the Normal Basis Theorem

The Normal Basis Theorem: If $E/F$ is a finite Galois extension, then there exists $a \in E$ such that the orbit of $a$ under the action of $\mathrm{Gal}(E/F)$ is a basis for $E$ as a vector space ...
11
votes
0answers
839 views

Galois theory timeline (II)

This question is a sequel. I structured the previous one around Emil Artin's classic treatment of Galois theory from the 1940s, though making clear some reservations of my own about whether Artin ...
15
votes
2answers
2k views

Galois theory timeline

A recent question on the history of Galois theory wasn't the most satisfactory. But the historical issues do seem quite attractive. They relate to innovation, and to exposition. There is a perspective ...
6
votes
3answers
3k views

What was Galois theory like before Emil Artin?

I read that the primitive element theorem for fields was fundamental in expositions of Galois theory before Emil Artin reformulated the subject. What are the differences between pre and post-Artin ...