Tagged Questions

6
votes
3answers
128 views

Preparations for reading Algebraic Number Theory by Serge Lang

I am eager to learn algebraic number theory. It seems that Serge Lang's Algebraic Number Theory is one of the standard introductory texts (correct me if this is an inaccurate assessment). I flipped ...
11
votes
1answer
93 views

Introduction to the trace formula for people outside number theory

I am looking for references on the trace formula, by which I mean the Selberg trace formula and its successor the Arthur-Selberg trace formula. I am aware that there are "standard references" on the ...
4
votes
3answers
75 views

Proof of Hasse-Minkowski over Number Field

Can anyone point me towards a good reference which contains the proof of the Hasse-Minkowski theorem of quadratic forms over a number field? Serre's "A Course in Arithmetic" has a self contained proof ...
5
votes
4answers
151 views

Reference request for Algebraic Number Theory sources for self-study

I would appreciate any suggestions for book or notes on ANT at a level that I would characterize as advanced beginner. I.e., something assuming familiarity with topics in Dummit & Foote, that is a ...
0
votes
0answers
27 views

A trace formula for an extension of complete valued fields

I'm trying to understand the following fact. Proposition. Let $L/K$ be a finite separable extension of complete valued fields, with respective valuations $w$ and $v$, valuation rings $\mathcal{O}_L$ ...
0
votes
0answers
64 views

How can commuting with Frobenius imply the order of an element in the inertia group.

In this video, one asserts, in the beginning, that, for $\tau\in \mathbb V_0$ such that $\tau$ generates $V_0/V_1$ in the quotient group, and $\sigma\in \mathbb Z$ which is a Frobenius in $\mathbb ...
1
vote
0answers
65 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 ...
2
votes
0answers
107 views

A book suggestion on algebraic number theory

I'm looking for a book on Algebraic Number Theory, which is somewhat in Analytic spirit. In particular, I want to see the precise connection between $$\delta_{f}(p)=\{a\pmod p : f(a) \equiv 0 ...
2
votes
2answers
192 views

Book recommendations for commutative algebra and algebraic number theory

Are there any books which teach commutative algebra and algebraic number theory at the same time. Many commutative algebra books contain few chapters on algebraic number theory at end. But I don't ...
3
votes
0answers
34 views

Schneider's theorem about the transcendence of values of the $j$-function

It is known that the $j$-function takes algebraic values when evaluated at imaginary quadratic integers. This is a result that was proved by Schneider in 1937 apparently. To be precise, Schneider ...
4
votes
1answer
103 views

Group of finite ideles

Let $K$ be a number field. Let $\mathbb{I}_f$ denote the group of finite ideles, and let $\phi: K^{\times} \rightarrow \mathbb{I}_f$ be the diagonal embedding. On page 167 of his notes on Class Field ...
2
votes
1answer
207 views

How many solutions for $x^2 = 1$?

Let $F$ be an non-archimedean local field, let $o$ be its ring of integers, and let $p$ be the maximal ideal Is there a closed form for the cardinality $$ | \{ x \in o / p^N: x^2 = -1 \bmod p^N\} | ...
4
votes
1answer
269 views

Special privilege enjoyed by Elliptic Curves with Complex Multiplication

I think after reading the title one may understand the intention of me, this question is concerned about the Elliptic curves having a Complex Multiplication. I have been reading many theorems, ( ...
0
votes
1answer
67 views

Generalizations of Monogenic Fields

We recall that a number field $K$ is called monogenic if there exists some $\alpha \in \mathcal{O}_K$ such that $\mathcal{O}_K = \mathbb{Z}[\alpha]$. If instead we take a tower of finite extensions ...
0
votes
1answer
119 views

Reference for proof of “Dedekind's Criterion”?

It was mentioned to me briefly in passing about a criterion for rings of integers, referred to as Dedekind's Criterion. The Criterion essentially said that a ring $\mathbb{Z}[\omega]$ (for ...
2
votes
2answers
223 views

Tamagawa numbers and Genus class numbers

I was reading the paper of Prof.Franz Lemmermeyer titled "Pell-conics" which is here, in that the author writes in page 9 that one can define Tamagawa numbers as $$ c_p = \begin{cases} 2 & \text{ ...
0
votes
1answer
164 views

A reference and an explanation needed?

In my previous question I was asking for a method to construct a global point if we have local points with us which is here, but I got an answer, it didn't serve the entire purpose, but later on due ...
7
votes
2answers
206 views

Good undergraduate level book on Cyclotomic fields

I have Lang's 2 volume set on "Cyclotomic fields", and Washington's "Introduction to Cyclotomic Fields", but I feel I need something more elementary. Maybe I need to read some more on algebraic number ...
15
votes
1answer
491 views

Brave New Number Theory

I suppose this is an extremely general question, so I apologize, and perhaps it should be deleted. On the other hand it's an awesome question. Is it clear exactly how much (assumedly algebraic) ...
2
votes
1answer
89 views

Simple Answer to Showing $N_{k/\mathbb{Q}}(\alpha)=N((\alpha))$

So I'm trying to show that if we have some number field $k/\mathbb{Q}$ and ring of integers $R_k\subset k$, and an element of $R_k$, say $\alpha$, that the field norm of $\alpha$ is equal to the ...
6
votes
1answer
332 views

What does Tate mean when he wrote “Higher dimensional class field theory” in the new preface to the Artin-Tate book and another question?

This is probably well-known to the experts or many number theory students, but since I am just starting to learn class field theory (with some basic knowledge of algebraic numbers, e.g. the 3 basic ...
6
votes
2answers
242 views

The Néron-Tate canonical height on elliptic curves

I have been trying to understand the Néron-Tate global canonical height of algebraic points on elliptic curves. Let $K$ be a number field, $E$ an elliptic curve (over $\mathbb{Q}$, say), and $E(K)$ ...
5
votes
1answer
309 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 ...
27
votes
4answers
1k 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 ...
1
vote
0answers
86 views

Hermite normal form and saturation

Recall that if $M$ is a submodule of $\mathbb{Z}^n$, then the saturation of $M$ (in $\mathbb{Q}$) is defined to be $\mathbb{Z}^n \cap (\mathbb{Q}\otimes_{\mathbb{Z}} M)$. According to an article of ...
5
votes
1answer
454 views

How many real quadratic number fields have the class number 1?

I know that in general the number of ideal classes are not 1, and that there are only 9 imaginary quadratic number fields which are principal ideal domains, i.e. $\mathbb(Q(\sqrt{-m}))$ where m is 1, ...
1
vote
2answers
169 views

Where can I find Heegner's proof?

Where can I read a corrected up to date version of Heegner's solution of the class 1 problem of Gauss?
5
votes
1answer
258 views

Does every ideal class contains a prime ideal that splits?

Suppose you have a number field $L$, and a non-zero ideal $I$ of the ring of integers $O$ of $L$. Question part A: Is there prime ideal $\mathcal{P} \subseteq O$ in the ideal class of $I$ such that ...
5
votes
1answer
256 views

Any resource of the applications of the theory of class fields

We all agree that the theory of class fields plays an eminent role in modern number theory. Nevertheless, what was our main concern is that how to solve various Diophantine equations to which the ...
3
votes
0answers
92 views

Where can the original paper by Takagi in English be found?

As we all know, the theory of class fields began at the paper by Takagi on the abelian extensions of the field of rational numbers. Then one naturally has the following Where can one find the ...
1
vote
1answer
614 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 ...
4
votes
0answers
128 views

Is there a notion of *p-adic Dedekind Domains*?

As we all know, the ring $Z_p$ can be constructed as the projective limit of the rings $Z/p^{n}Z$. Now is there any generalization such as the p-adic completions of a Dedekind Domain? This might be ...
3
votes
0answers
92 views

places valuations and so on

I am looking for complete references on places, valuations and so on. In particular I need to understand the meaning of "a finite place does not divide a prime number $\ell$" and what is the Frobenius ...