4
votes
0answers
79 views

Any abstract algebra book with programming (homework) assignment?

All: I had studied abstract algebra long time ago. Now, I would like to review some material, particularly about Galois theory (and its application). Can anyone recommend an abstract algebra book ...
6
votes
2answers
63 views

Does the data of Galois group, ramified places, and inertia groups, determine a Galois number field?

Suppose I tell you that $K/\mathbb{Q}$ is a finite Galois extension, and I specify the Galois group $G$, and suppose further that I give you a finite list $S$ of places of $\mathbb{Q}$ and for each ...
6
votes
4answers
213 views

Learning about Grothendieck's Galois Theory.

I have background in category theory and I am familiar with the very basics of algebraic geometry - Chapters I and II of Hartshorne. What would be a recommended (self-contained, maybe?) text for ...
7
votes
1answer
117 views

Question about inverse Galois problem

I have a question... if for every finite simple group, we can construct a Galois extension over $\mathbb{Q}$ with that Galois group, does it follow that we can construct Galois extensions over ...
2
votes
1answer
39 views

Help with Polynomial Roots Problem

Let's consider the case of two variables, $p\in\mathbb{R}[x,y]$. Suppose I want to find when there is $c\in\mathbb{R}$ such that $$p(x,x)+p(x,c-x)-p(c-x,x)-p(c-x,c-x)=0 \textbf{ for all } ...
4
votes
2answers
297 views

Proof of Fermats Last Theorem for Given Exponent

Where can I find reasonably short and elementary proofs (using basic concepts of ring, field, galois theory) of Fermat's Last Theorem for specific n? For example, $n=5,7,13$?
3
votes
1answer
219 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?
12
votes
1answer
230 views

Semialgebraic conditions that convey properties of Galois group

Let $f \in \mathbb{Z}[x]$ be a polynomial of degree $n$ with integer coefficients and let $G_f$ be the Galois group of $f$ over $\mathbb{Q}$. I am trying to collect results that convey some ...
3
votes
1answer
384 views

Where to start with the Inverse Galois Problem

I have always been interested in the Inverse Galois Problem since I read the story of Évariste Galois. I have recently finished up courses in rings, fields, groups, and finally Galois theory. I really ...
5
votes
0answers
87 views

Classes of groups known to be realizable (IGP)

A finite group $G$ of order $n$ is said to be realizable (over $\mathbb{Q}$) if there exists a Galois extension $L/\mathbb{Q}$ such that $\mathrm{Gal}(L/\mathbb{Q})=G$. I'm curious what classes of ...
3
votes
2answers
473 views

source to learn Galois Theory

What kind of recommendations do you have for a very good source to learn Galois Theory? Is there any Atiyah-MacDonald-type book on Galois theory? What is your opinion on the chapters from Lang and ...
2
votes
0answers
41 views

presentation of the inertia group of $p$-adic fields

It is known (see here) that the absolute Galois group of a $p$-adic number field $K$ (ie a finite extension of $\mathbb Q_p$) is topologically finitely presented for any odd prime $p$. Is it also ...
7
votes
3answers
959 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 ...
7
votes
1answer
310 views

Maximal ideals in polynomial rings over a field

Let $K$ be an algebraically closed field and let $k$ be a subfield of $K$ such that the field extension $K \mid k$ is algebraic. Let $B$ be the polynomial ring $K [x_1, \ldots, x_n]$ and let $A$ be ...
3
votes
1answer
78 views

When are powers of primitive elements still primitive elements

This question is motivated by this question and is tangentially related to this question. Let $L/K$ be a finite Galois extension of fields. Pick $\alpha \in L \setminus K$ and consider the simple ...
0
votes
1answer
339 views

On Galois Theorem and Dummit and Foote text !

in dummit and foote text , galois theory is presented in chapter 14 . group theory is presented in 6 chapter , ring theory in 3 chapter and so on my qestion is , which chapters of the text is ...
4
votes
4answers
236 views

Necessarily complex analytic proofs in algebra.

Does anyone know of an example where complex analysis is necessary to prove something in algebra? I would be particularly interested in results from group theory or Galois theory. In an ideal ...
4
votes
1answer
367 views

Inverse Galois problem for small groups

I am looking for a list of all small groups (maybe order $\leq 20$) realized as the Galois groups of a polynomial over $\mathbb{Q}$, with proof. Any idea where I could find these? Partial answers or ...
25
votes
1answer
345 views

Is the Galois group associated to a random polynomial solvable with probability 0?

Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. Let $r_1,\ldots,r_n$ be the roots of $P$ and consider ...
2
votes
1answer
146 views

Automorphism extension property of Galois extensions

If $L$ is a Galois extension of $K$ and $M$ is a finite Galois subextension of $L \mid K$, then a standard lemma says that any automorphism of $M \mid K$ can be extended to an automorphism of $L \mid ...
2
votes
0answers
236 views

Serre's Topics in Galois Theory

I am supposed to write a running notes on Hilbert Irreducibility Theorem from Serre's Topics in Galois Theory as part of my master's Algebra course work. But I don't have any knowledge of Algebraic ...
2
votes
1answer
191 views

Transition between field representation

In a number of papers related to efficient implementation of the AES Sbox, people are computing stuff (the multiplicative inverse for instance) in GF(($2^4$)$^2$) instead of GF($2^8$). In some cases ...
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.
10
votes
1answer
404 views

Reference request: Abel or Ruffini's proof of the Abel-Ruffini theorem

The standard proof of the Abel-Ruffini theorem that people learn is based on Galois theory and the notion of a solvable group, but my understanding is that the original proof predates Galois theory. ...
20
votes
2answers
1k views

Original works of great mathematician Évariste Galois

Through this question I wanted to know the original works of Galois. When I was reading Galois theory ( since from last month ) , I have been seeing one common line in every book, whose essence ...
7
votes
0answers
472 views

“The Galois group of $\pi$ is $\mathbb{Z}$”

Last year, in a talk of Michel Waldschmidt's, I remember hearing a statement along the lines of the title of this question: The Galois group of $\pi$ is $\mathbb{Z}$. In what sense/framework is ...
4
votes
1answer
445 views

What is a Weil-Deligne representation?

Can anyone explain (or give reference) Weil-Deligne representations, and how they are linked with Galois representations of p-adic field (on l-adic vector spaces or else) ?
11
votes
3answers
1k views

Exercises on Galois Theory

I need a source for exercises on classical Galois Theory, or to be more specific, Galois extensions of finite fields and the rationals as well as applications (solvability by radicals, for example). ...
8
votes
1answer
646 views

Reference book for Artin-Schreier Theory

The aim of the question is very simple, I would like to study Artin-Schreier Theory, but I have had embarassing difficulties in finding a book which could help me in doing that. In specific I'm ...
3
votes
1answer
122 views

Does there exist an English translation of Distler's paper on solving polynomials by radicals?

The Radiroot package for GAP by Andreas Distler solves by radicals polynomials with solvable Galois group. The package uses the algorithm described in Distler's paper "Ein Algorithmus zum Lösen einer ...
12
votes
1answer
376 views

Transcendental Galois Theory

Is there a good reference on transcendental Galois Theory? More precisely, if $K/k$ admits a separating transcendence basis (or maybe if it is a separably generated extension) it seems to me that ...
3
votes
1answer
215 views

Where can I find the paper by Shafarevich on the result of the realization of solvable groups as Galois groups over $\mathbb{Q}$?

I ha come across a book on groups as Galois groups, and in the introduction it mentions the paper by I.R. Shafarevich which says that every solvable group can be realized as Galois groups of some ...