5
votes
1answer
35 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 ...
5
votes
1answer
73 views

Gentle introduction to algebraic number theory

For context, I am a undergraduate majoring in math. I've taken two semesters of algebra (though I am still a bit shaky on Galois theory). I just finshed a course in elementary number theory which used ...
5
votes
0answers
58 views

Literature to the ring $\mathbb{Z}[\phi]$ where $\phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio

I know few about algebraic number theory but recently I stumbled upon the ring $\mathbb{Z}[\phi]$ where $\phi = \frac{1+\sqrt{5}}{2}$ is the golden ratio. It seems to be a very interesting object to ...
1
vote
1answer
37 views

What are the roots of unity quadratic integers?

The article of Roots of unity in wikipedia implies that the following roots of unity are quadratic numbers: $$ \{\pm 1\}, \{\pm 1,\pm i\}, \{\pm 1,\pm \zeta,\pm \zeta^2\}. $$ where $\zeta=\exp(2\pi ...
1
vote
0answers
50 views

Generalizations of results on the sum of divisors function over $\mathbb{Q}$ to number fields

Consider the sum of divisor function $$ \sigma_0(n) = \sum_{d\mid n} 1. $$ This is known to satisfy $\sum_{n\leq x} \sigma_0(n) = (x\log x)+2\gamma x+\mathcal{O}(\sqrt{x})$. If, instead, we shift the ...
0
votes
0answers
25 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., ...
3
votes
2answers
69 views

Where can i find resourses to study this algebraic number theory?

Where can i find material to study depper the farey fractions (continued fractions)? I triying to solve problems like these: 1.- Show that two consecutive convergent at least one of them satisfy: ...
1
vote
0answers
29 views

integral basis for an arbitrary cubic Galois field

I wonder where I can find some information (possible a book) about finding an integral basis for cubic Galois fields? I know that for pure cubic fields there exists a simple criterion according to ...
3
votes
4answers
137 views

Textbook for graduate number theory

I am attending a graduate number theory, the professor did not assign any textbook. The materials are somewhere along the advanced/algebraic level such as Ring of Gaussian Integers, Quadratic Number ...
3
votes
0answers
49 views

class field theory via schemes?

I know there is a close relationship between algebraic number theory and algebraic geometry. And in particular the theory of schemes is of many uses in algebraic number theory. Since I think the peak ...
7
votes
0answers
150 views

Adelic/Idelic method for number fields

I'm searching for a place where is developed all the machinery of adeles and ideles of a number field; the functional equation for the zeta functions is gained by idelic integration, and it's seen the ...
6
votes
1answer
172 views

What is the best book learn Galois Theory if I am planning to do number theory in future?

What is the best book learn Galois Theory if I am planning to do number theory in future? In a year i'll be joining for my Phd and my area of interest is number theory. So I want to know if there is ...
0
votes
1answer
91 views

Power series ring over a ring of integers

Let $K/\mathbb {Q}_p$ be a finite extension, $\mathcal{O} := \mathcal{O}_K$ the ring of integers of $K,$ $\frak p$ the maximal ideal of $\mathcal{O}$, and $\pi$ a uniformizer, i.e., $\frak{p} = ...
3
votes
2answers
130 views

Introductive Book on Modular Forms

I'm looking for an introductive book on Modular Forms and their applications to Algebraic Geometry and Algebraic Number Theory. Some Ideas? Explaining you my prerequisites, I've a good knowledge of ...
2
votes
2answers
64 views

Total ramification in infinite Galois extension

Let $L/K$ be an infinite galois extension of fields, and $\frak p$ a prime of $K$ and $\frak P$ a prime of $L$ above $\frak p.$ We define the ramification index of $\frak p$ by $e(\frak ...
1
vote
0answers
29 views

Examples of idelic Artin map

I do not know of any source on class field theory which explains the idelic Artin map through a couple of examples. Please provide me some.
12
votes
2answers
310 views

Are there Groups of Strictly Primes

Motivation Since Euclid's proof of the infinitude of the primes, the structure and properties of primes has always fascinated mathematicians. This lead to great work in their properties and ...
5
votes
1answer
101 views

The number of Number Fields of discriminant less than or equal to a particular value

By Hermite's theorem, there are only finitely many number fields of bounded (equivalently, fixed) discriminant. But I assume that people have collected data about this to say more than just that? I ...
1
vote
1answer
63 views

Mathematica code as reference?

I am preparing a paper wherethere are a lot of algebraic simplifications needed. One of them takes 3 pages to get the result. Instead of consuming all the pages can I cite a mathematica code which is ...
2
votes
1answer
62 views

Automorphism action on Artin map

I would like to refer you to Serge Lang's book on Elliptic Functions $\S10.4$ (page 140). I'm trying to understand a part in theorem 10 where he proves that $$\mu(\mathfrak{p})'=\mu(\mathfrak{p}')$$ ...
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 ...
2
votes
0answers
69 views

Kronecker's approach to unique factorisation in algebraic number theory: books and references

I have done a short course (one semester) on algebraic number theory at beginning graduate level, in which the Dedekind theory of ideals features prominently. However I have since discovered that ...
8
votes
2answers
118 views

Reference for relation between class number of $\Bbb Q[\sqrt{-p}]$ and partial quotients of $\sqrt p$

So in Ireland and Rosen's, "Classical Introduction to Modern Number Theory", they mention the following incredible fact at the end of Chapter 13, section 1. Suppose $p \neq 3$ and $p \equiv 3 \pmod 4$ ...
9
votes
1answer
666 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 ...
2
votes
1answer
34 views

What is a place?

In Specialization of Quadratic and Symmetric Bilinear Forms (page: 3) the author writes "Let also $\lambda: K \to L \cup \infty$ be a place, $\mathfrak o = \mathfrak o_\lambda$ the valuation ring ...
5
votes
1answer
131 views

What is the integral homology of $\mathrm{GL}_2(\mathbb{Z}[i])$?

I am currently trying to compute homology groups of general linear groups over the ring of integers of an imaginary quadratic number field. As I would like to check my results I would like to know if ...
7
votes
3answers
575 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 ...
12
votes
1answer
191 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 ...
5
votes
3answers
206 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 ...
6
votes
4answers
328 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 ...
1
vote
0answers
45 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
70 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
1answer
132 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
180 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 ...
3
votes
2answers
385 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
60 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 ...
3
votes
3answers
1k views

Pre-requisites needed for algebraic number theory

I acknowledge my limited knowledge of abstract algebra(My background comprising groups and subgroups from Herstein's Topics in Algebra is hardly worth mentioning) .And yet, I confess I really like ...
4
votes
1answer
124 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
222 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\} | ...
7
votes
1answer
346 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
84 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
181 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
245 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
171 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 ...
8
votes
2answers
261 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 ...
19
votes
1answer
692 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
106 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 ...
12
votes
4answers
1k views

Beginner's text for Algebraic Number Theory

What's good book for learning Algebraic Number Theory with minimum prerequisites? Assume that the reader has done an basic abstract algebra course.
10
votes
1answer
483 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
330 views

The Néron-Tate canonical height on elliptic curves

I have been trying to understand the NĂ©ron-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 $E(K)$ ...