Questions related to the algebraic structure of algebraic integers

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13
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1answer
326 views

2-Torsion Group Scheme

Consider the elliptic curve $zy^2 + z^2y = x^3.$ I would like to explictly compute the 2-torsion group scheme, $E[2],$ over $\mathbf{Spec}(\mathbb{Z}_2),$ but I'm having a tough time writing down the ...
13
votes
1answer
207 views

Modular interpretation of modular curves

Let $\Gamma$ be a congruence subgroup of level $N$. What is the modular interpretation of $\Gamma\backslash \mathcal{H}^*$, by that I mean what are the elliptic curves + additional structure it ...
13
votes
1answer
181 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 ...
13
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0answers
1k views

What are some strong algebraic number theory PhD programs?

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, ...
13
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0answers
183 views

Homogeneous forms of degree $n$ in $n$ indeterminates over $\mathbb{Z}$: which ones come from the norm of a number field?

Is there a characterization of the homogeneous forms of degree $n$ in $n$ indeterminates over $\mathbb{Z}$ which occur as the norm of some algebraic number ring with a suitable $\mathbb{Z}$-basis? ...
12
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5answers
531 views

How does a Class group measure the failure of Unique factorization?

I have been stuck with a severe problem from last few days. I have developed some intuition for my-self in understanding the class group, but I lost the track of it in my brain. So I am now facing a ...
12
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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.
12
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2answers
583 views

How are the Tate-Shafarevich group and class group supposed to be cognates?

How can one consider the Tate-Shafarevich group and class group of a field to be analogues? I have heard many authors and even many expository papers saying so, class group as far as I know is ...
12
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2answers
316 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 ...
12
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1answer
148 views

Galois Properties of the Values of Modular Forms

Let $f \in S_0(N)$ be a normalized Hecke eigenform. It is well known that its coefficients are algebraic integers, and $f^\sigma$ lies in $S_0(N)$ for $\sigma \in G_{\mathbb{Q}}$. At CM points $z \in ...
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 ...
12
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2answers
555 views

A weak converse of $AB=BA\implies e^Ae^B=e^Be^A$ from “Topics in Matrix Analysis” for matrices of algebraic numbers.

It is a well known fact that if $A,B\in M_{n\times n}(\mathbb C)$ and $AB=BA$, then $e^Ae^B=e^Be^A.$ The converse does not hold. Horn and Johnson give the following example in their Topics in Matrix ...
12
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1answer
125 views

When is a number in $\mathbb{Q}(\sqrt{a_1},\ldots,\sqrt{a_n})$?

Given an algebraic number $\alpha$ with minimal polynomial $P(x)$ of degree $2^n$, how can I decide if there are integers $a_1,\ldots,a_n$ such that ...
12
votes
1answer
249 views

primes represented integrally by a homogeneous cubic form

Expired by this question Show determinant of matrix is non-zero I am moved to ask: Given integers $a,b,c,$ and cubic form $$ f(a,b,c) = a^3+2 b^3-6 a b c+4 c^3 = \left|\begin{bmatrix} a & 2c ...
12
votes
2answers
178 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 ...
11
votes
5answers
1k views

How to tell if a Fibonacci number has an even or odd index

Given only $F_n$, that is the $n$th term of the Fibonacci sequence, how can you tell if $n \equiv 1 \mod 2$ or $n \equiv 0 \mod 2$? I know you can use the Pisano period, however if $n \equiv 1$ or ...
11
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3answers
352 views

What is the quotient $\mathbb Z[\sqrt{3}]/(1+2\sqrt{3})$?

I am currently doing a past paper and it asks the following: Prove that for $I=(1+2\sqrt{3})$ we have $\mathbb Z[\sqrt{3}]/I$ a field with $11$ elements. If I assume standard algebraic number ...
11
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3answers
1k views

What are the units of cyclotomic integers?

This question made me realize I had a misconception about the cyclotomic integers: I thought the units were exactly the roots of unity. There are only finitely many units but infinitely many integers ...
11
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1answer
248 views

If $u=\frac{1+\sqrt5}{2}$, then $u^3=2+\sqrt5$, but $u^2=\frac{3+\sqrt5}{2}$. What is the group that measures the power that makes units look nice?

For $A=\mathbb{Z}[x]/(f)$ with quotient field $K$ and ring of integers $B$, does $U(B)/U(A)$ have a name? For instance $u = \tfrac{1+\sqrt{5}}{2}$ is a unit in $\mathbb{Q}[\sqrt{5}]$, but neither ...
10
votes
3answers
1k views

How many elements in a number field of a given norm?

Let $K$ be a number field, with ring of integers $\mathcal{O}_k$. For $x\in \mathcal{O}_K$, let $f(x) = |N_{K/\mathbb{Q}}(x)|$, the (usual) absolute value of the norm of $x$ over $\mathbb{Q}$. ...
10
votes
2answers
388 views

Factoring a number of complex integers?

Say you are given a number (ex: $377$) and you express it in a form that allows you to factor it over the complex integers: Notice, $377 = 16^2 + 11^2$ Thus: $(16 + 11i) $ and $(16 - 11i)$ Are ...
10
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2answers
275 views

Ring of integers is a PID but not a Euclidean domain

I have noticed that to prove fields like $\mathbb{Q}(i)$ and $\mathbb{Q}(e^{\frac{2\pi i}{3}})$ have class number one, we show they are Euclidean domains by tessalating the complex plane with the ...
10
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2answers
1k views

Relationship between Cyclotomic and Quadratic fields

Since $\varphi(p)=p-1$ is even the p'th cyclotomic field contains some quadratic field. Hecke says that in fact every quadratic field is contained by some cyclotomic field. What is this theorem ...
10
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3answers
608 views

What is so special about negative numbers $m$, $\mathbb{Z}[\sqrt{m}]$?

This question is based on a homework exercise: "Let $m$ be a negative, square-free integer with at least two prime factors. Show that $\mathbb{Z}[\sqrt{m}]$ is not a PID." In an aside comment in the ...
10
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2answers
476 views

Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$

1) Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$. Prove: Is it true that $a \in \mathbb{Q}$? 2) Suppose $a \in \mathbb{C}$, ...
10
votes
1answer
239 views

What does the German word “Zerlegungsautomorphismus” translate to?

I would like to know if any of our German friends can translate that word for me? Zerlegung is factorisation isn't it? So what is factorisation automorphism? This is taken from Deuring's paper “Die ...
10
votes
2answers
823 views

Ideal class group of a one-dimensional Noetherian domain

Let $A$ be a one-dimensional Noetherian domain. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated $A$-module. It is well-known that B ...
10
votes
1answer
240 views

Tate's Thesis: in what sense is Tate's Theorem 4.2.1 the Riemann-Roch theorem for curves?

I am reading Tate's Thesis. Tate derives a theorem which he calls "the number-theoretic analogue of the Riemann-Roch theorem" from an abstract Poisson summation formula. I am accustomed to thinking of ...
10
votes
3answers
194 views

Does $\mathbb{F}_p((X))$ has only finitely many extension of a given degree?

We know that $\mathbb{Q}_p$ has only finitely many extensions of a given degree in its algebraic closure. Is it the same for $\mathbb{F}_p((X))$?
10
votes
1answer
493 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 ...
10
votes
2answers
291 views

Classifying splittings of primes?

I was wondering what general strategies are available to figure out if a prime splits? I know for quadratic extensions there aren't too many possibilities for how a prime can split, so we essentially ...
10
votes
1answer
152 views

Elementary proof that $\mathfrak{p}$ unramified in $L,L'$ implies unramified in $LL'$?

Let $K$ be a number field, let $\mathfrak{p}$ be a prime of $K$, and let $L,L'$ be extensions of $K$. Suppose $\mathfrak{p}$ doesn't ramify in either $L$ or $L'$. Is there a simple proof that ...
10
votes
2answers
423 views

Eisenstein and Quadratic Reciprocity as a consequence of Artin Reciprocity, and Composition of Reciprocity Laws

Question 1: I've heard that Eisenstein and Quadratic Reciprocity can be derived from the Artin Reciprocity by applying it to certain field extensions. But I haven't seen on any reference an explicit ...
10
votes
1answer
305 views

How to determine all valuations of the field $\mathbb{Q(\sqrt[n]{2})}$?

This is an exercise from the book Algebraic Number Theory by Jurgen Neukirch, on page 166. And, after solving several previous exercises, I found this to be particularly difficult to solve. I am ...
10
votes
4answers
169 views

Is the number of irreducibles in any number field infinite?

Are there infinitely many irreducibles in the ring of integers of any algebraic number field ? I tried to follow the same argument as we usually do for integers. Suppose there are finitely many ...
10
votes
1answer
196 views

Galois representations and normal bases

I am not very familiar with the theory of Galois representations, but I do know a bit about both Galois theory and representation theory. Recently I learned about the notion of a normal basis for a ...
10
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2answers
467 views

Cubic polynomials that generate the same extension?

For quadratic extensions we can easily determine when $\mathbb{Q}(\sqrt{a})=\mathbb{Q}(\sqrt{b})$ by checking if $a/b$ is a square and this is easy to prove. I was wondering if there are any good ...
10
votes
1answer
215 views

Connection between ramification in number fields and Clifford theory

Consider algebraic number fields $\mathbb{Q} \subseteq K \subseteq L$ with rings of integers $\mathbb{Z}\subseteq \mathcal{O}_K \subseteq \mathcal{O}_L$. If $0 \neq \mathfrak{p} \trianglelefteq ...
10
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1answer
285 views

What are the branches of the $p$-adic zeta function?

I'm reading the book $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz. In it, Koblitz wants to iterpolate the Riemann Zeta function for the values $\zeta_p(1-k)$ with $k \in ...
10
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0answers
153 views

CFT via Brauer groups vs via ideles

I am interested in the relationship between the following two versions of CFT: Version 1: (Brauer Group Version) Let $K$ be a number field. One constructs, for every finite place $v$ of $K$, a map ...
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0answers
403 views

extension of Euler's totient function to number fields

It is well known that the Euler totient function $\varphi$ satisfies the formula $n = \sum_{d | n}\varphi(d)$. This follows for example from the fact that $\mathbb Z / n \mathbb Z$ can be written (as ...
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5answers
495 views

Motivation behind the definition of ideal class group

Let $O$ be a Dedekind domain and $K$ its field of fractions. The set of all fractional ideals of $K$ form a group, the ideal group $J_K$ of K. The fractional principal ideals $(a) = aO, a \in K^*$, ...
9
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5answers
215 views

Are numbers of the form $n^2+n+17$ always prime

Someone claimed that a number, multiplied by the number after it plus 17 is always prime, and showed several cases. I'm not a complete amateur in Number Theory, and I know that $17*18+17=17*19$, so it ...
9
votes
1answer
238 views

Why is $(\sqrt{2}+\sqrt{3})^{2008}$ so close to an integer?

Using 5000-digit precision in PARI/GP, I discovered that the fractional part of $(\sqrt{2}+\sqrt{3})^{2008}$ is extremely small, less than $10^{-999}$. Is there a simple explanation for this fact ? ...
9
votes
2answers
212 views

Groups of units of $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$

On page 230 of Dummit and Foote's Abstract Algebra, they say: the units of $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$ are determined by the integers $a,b$ with $a^2+ab+b^2=\pm1$ i.e. with ...
9
votes
4answers
859 views

(Simple?) applications of Class Field Theory?

Does anyone know any simple/nice applications of class field theory? I would really like to find one related to diophantine equations, but anything you got would be good. Thanks
9
votes
2answers
360 views

In what senses are archimedean places infinite?

According to Bjorn Poonen's notes here (§2.6), we should add the archimedean places of a number field $K$ to $\operatorname{Spec} \mathscr{O}_K$ in order to get a good analogy with smooth projective ...
9
votes
1answer
887 views

Euler's remarkable prime-producing polynomial and quadratic UFDs

Good example of a polynomial which produces a finite number of primes is: $$x^{2}+x+41$$ which produces primes for every integer $ 0 \leq x \leq 39$. In a paper H. Stark proves the following result: ...
9
votes
2answers
888 views

Ramification in a tower of extensions

I'm trying to make sense of all these theorems related to ramification. I was hoping someone would summarize these results. Assume we have: An extension $L/K$ and a some subextensions ...
9
votes
2answers
282 views

Are rational numbers + numbers constructed with a root = algebraic numbers?

I'm a math newbie, so an intuitive explanation is the most helpful for me, but don't pull your punches with the formulas, if you feel like it. We can construct the rational numbers using the division ...