2
votes
2answers
90 views

Solved Problems in Algebraic Number Theory

I'm not too sure whether this is the right place to ask this (and please correct me if it is not), but I'm currently studying a course in Algebraic Number Theory and would like to be pointed in the ...
0
votes
0answers
23 views

Theory of irrationalities- Faddeev's book

Does anyone know where (if available) I can get a free access to Delone, B. N., Faddeev, D. K., ''The theory of irrationalities of the third degree'' Transl. Math. Monographs 10, Amer. Math. Soc., ...
2
votes
1answer
81 views

How to proceed doing number theory?

I'm an undergrad majoring in mathematics. Being in first year I'm still exploring new branches of mathematics and till now, It is analysis and Number theory that I've come to have a great interest ...
4
votes
1answer
260 views

Forcing the discriminant of an integral basis to be a Carmichael number.

I was thinking about the following lemma recently. Lemma: Let $K=\mathbb{Q}(\theta)$ for some algebraic number $\theta$ and let $n=[K:\mathbb{Q}]$. If $\{\tau_1, \,\dots\,, \tau_n\}$ consists of ...
8
votes
2answers
68 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 ...
13
votes
1answer
169 views

What's so cool about local compactness?

As I study more algebraic number theory, I hear more and more often about local compactness: locally compact fields, locally compact topological groups, Stone-Čech compactification of locally compact ...
4
votes
1answer
99 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 ...
10
votes
0answers
612 views

What are some strong PhD programs in (algebraic) number theory?

I am currently applying for PhD programs in the US. My main interests are number theory and algebra. More specifically, I am interested in algebraic number theory (number fields, Galois groups, ...
12
votes
2answers
153 views

Are there applications of noncommutative geometry to number theory?

The marriage of algebraic geometry and number theory was celebrated in the twentieth century by the school of Grothendieck. As a consequence, number theory has been completely transformed. On the ...
6
votes
1answer
89 views

Tate's Thesis: Meaning of Local Functional Equation

I am studying the development of Tate's Thesis in Lang's Algebraic Number Theory and have a conceptual question. The setting: Let $k=\mathbb{Q}_p$. Let $\mu$ be the unique Haar measure giving ...
18
votes
1answer
402 views

Why did Gauss think the reciprocity law so important in number theory?

Gauss's Disquitiones Arithmeticae centers around the quadratic reciprocity law. It seems that he developed the genus theory of integral binary quadratic forms to find a natural proof of the quadratic ...
9
votes
1answer
524 views

Special cases of the Stark-Heegner theorem with simple proofs

The Stark-Heegner theorem states that the ring of integers of the quadratic number field $\mathbb Q(\sqrt{m})$, where $m$ is a squarefree negative integer, is a principal ideal domain, iff ...
3
votes
2answers
84 views

Why do we only consider quadratic domains as Euclidean domains with squarefree integers?

I have been reading "Introductory algebraic number theory" by Alaca and Williams, and in the opening chapters they use the quadratic domains $\mathbb{Z}+\mathbb{z}(\sqrt{m})$ for non-square $m$ and ...
3
votes
1answer
105 views

bests book of representation theory for algebraic number theorists

I am looking for some of the best books on representation theory for an algebraic number theorists> I would prefer a book that is more number theoretical (e.g, galois representations, p adic ...
40
votes
3answers
926 views
5
votes
2answers
179 views

Are algebraic numbers analogous to group elements with finite order?

Would you say that the "elements with finite order" in group theory are analogous to "algebraic numbers" in field theory? I thought this is the case since requiring an algebraic number $\alpha$ to be ...
1
vote
1answer
126 views

Applications of prime-number theorem in algebraic number theory?

Dirichlet arithmetic progression theorem, or more generally, Chabotarev density theorem, has applications to algebraic number theory, especially in class-field theory. Since we might think of the ...
7
votes
2answers
350 views

Classgroup of $\mathbb{Q}(\sqrt{2},\sqrt{-13})$

How would you compute the classgroup of the biquadratic number field $\mathbb{Q}(\sqrt{2},\sqrt{-13})$? I would prefer a method as "from scratch" as possible. Please avoid, if possible, quoting ...
4
votes
1answer
182 views

Are Different Areas of Number Theory That Different

I was wondering if number theorists are "number theorists," or eventually resolve themselves into one of the various branches - i.e., algebraic, analytic, etc. Also out of curiosity, I was wondering ...
18
votes
5answers
657 views

Elementary results from Algebraic Number Theory

The purpose of this question is to motivate me to study algebraic number theory. Let me explain. My motivation for studying number theory is to learn about beautiful results with simple, accessible ...
5
votes
2answers
218 views

Group actions in towers of Galois extensions

Assume we are given an extension of number fields or $\mathfrak{p}$-adic number fields $L/E/K$ where each extension is abelian and $L/K$ is only assumed Galois. Now take any element $\sigma\in ...
1
vote
2answers
95 views

Question about topology on $K^\times$ in local CFT

I'm trying to parse a page in Milne's CFT notes. The local reciprocity law gives us isomorphisms $$\phi_{L/K}:K^\times/Nm(L^\times)\to \textrm{Gal}(L/K)$$ for all abelian extensions $L$ of a ...
13
votes
2answers
351 views

Consequences of the Langlands program

I have been reading the book Fearless Symmetry by Ash and Gross.It talks about Langlands program, which it says is the conjecture that there is a correspondence between any Galois representation ...
31
votes
4answers
4k views

The Langlands program for beginners

Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things he must know before he can understand the Langlands program and its ...
2
votes
1answer
845 views

Are there any good algebraic geometry books to recommend? [duplicate]

Possible Duplicate: (undergraduate) Algebraic Geometry Textbook Recomendations I am interested in algebraic number theory and I am recently acquainted with the theory of valuations, which ...
7
votes
2answers
510 views

A question on FLT and Taniyama Shimura

Sometime back i watched the documentary of Andrew Wiles proving the Fermat's Last theorem. A truly inspiring video and i still watch it whenever i am in a depressed mood. There are certain ...