1
vote
0answers
78 views

Any other mathematicians like Galois in recent history?

Any other mathematicians like Galois in recent history ? For "someone like Galois", I mean someone who developed a completely new theory all by himself, solved a big problem and the theory has big ...
4
votes
1answer
81 views

What is the difference between field theory and Galois theory

I am about to finish the book Galois theory by Harold Edwards. I am planning to study Galois theory at a more advanced level or field theory. I am unable to decide because I don't know the difference ...
18
votes
3answers
327 views

Why there is much interest in the study of $\operatorname{Gal}\left(\overline{\mathbb Q}/\mathbb Q\right)$?

Let's start for a simple quote from wikipedia: "No direct description is known for the absolute Galois group of the rational numbers. In this case, it follows from Belyi's theorem that the ...
5
votes
0answers
104 views

Intuition about structures in Galois Theory.

Background Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like ...
4
votes
0answers
86 views

Representation Theory versus Galois Theory. [closed]

At my university there is a debate about whether it is better to require students to have taken a class in Galois theory or to require a class in representation theory for admission into the graduate ...
3
votes
1answer
221 views

Best book ever on Galois theory (and differential galois theory) [closed]

Which is the single best book for Galois theory (that includes differential Galois theory) that everyone who loves pure Mathematics should read?
8
votes
2answers
86 views

Conceptual reason why a quadratic field has $-1$ as a norm if and only if it is a subfield of a $\mathbb{Z}/4$ extension?

I have convinced myself in a computation-heavy, ad-hoc way that a quadratic extension $K$ of $\mathbb{Q}$ occurs as the unique quadratic subfield of a $\mathbb{Z}/4$-extension of $\mathbb{Q}$ if and ...
2
votes
1answer
90 views

Describing the impossibility of trisecting the angle to high school students.

Does anyone have an idea on whether it would be possible to present the proof of the impossibility of trisecting the angle (or doubling the cube, for example) in order to demonstrate the power of ...
4
votes
1answer
107 views

What is going on in this degree 8 number field that fails to be a quaternion extension of $\mathbb{Q}$?

This is a soft but very mathematically hands-on question. Hopefully it will be interesting to more than just me. Thanks in advance for your help in thinking clearly about what follows. I have been ...
25
votes
2answers
625 views

Intuitive reasoning why are quintics unsolvable

I know that quintics in general are unsolvable, whereas lower-degree equations are solvable and the formal explanation is very hard. I would like to have an intuitive reasoning of why it is so, ...
7
votes
3answers
966 views

Self teaching Galois Theory

At uni, I did a module in group theory which I really enjoyed. I also did one on number theory which had aspects of rings and fields in it and I enjoyed learning this. Now that I have finished uni, I ...
1
vote
0answers
98 views

Understanding Andrew Wiles proof of Fermat's last theorem [duplicate]

Possible Duplicate: knowledge needed to understand Fermat’s last theorem proof What are the prerequisites needed to understand wiles proof? Could someone sketch a roadmap of how it could be ...
3
votes
1answer
375 views

Tips for finding the Galois Group of a given polynomial

I am currently in an introductory Galois Theory course, and I thought it would be nice to compile a list of standard tricks for finding the Galois Groups of certain polynomials. I am studying from ...
6
votes
3answers
1k views

What is the prerequisite knowledge for learning Galois theory?

What is the prerequisite knowledge for learning Galois theory? I don't know what a ring is.
26
votes
10answers
2k views

Contributions of Galois Theory to Mathematics

What are the major and minor contributions of Galois Theory to Mathematics? I mean direct contributions (like being aplied as it appears in Algebra) or simply by serving as a model to other theories. ...