Questions related to the algebraic structure of algebraic integers

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15
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3answers
177 views

Given $d \equiv 5 \pmod {10}$, prove $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ never has unique factorization

With the exception of $d = 5$, which gives $\mathbb{Z}[\phi]$, of course (as was explained to me in another question). I'm not concerned about $d$ negative here, though that might provide a clue I ...
15
votes
1answer
442 views

Explicit automorphisms of the field of algebraic numbers

The field $\overline {\bf {Q}} $ of algebraic numbers admits many automorphisms other than conjugation. This follows from Galois theory: the field $\overline {\bf {Q}}$ can be realized as the union ...
15
votes
2answers
610 views

Formula for number of solutions to $x^4+y^4=1$, from Ireland and Rosen #8.18.

There is a sequence of three exercise in Ireland and Rosen's Introduction to Modern Number Theory, Chapter 8, page 106. I can do the first two, but can't finish the third. I can include the proofs to ...
15
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2answers
581 views

What is Riemann-Roch in arithmetic all about?

I learn number theory recently and I could not understand what Riemann-Roch was all about in arithmetic; could someone give me a bit hint? What is the advantage of viewing all this stuff geometrically ...
14
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2answers
532 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 ? ...
14
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5answers
748 views

An 'obvious' property of algebraic integers?

I am looking at the book A Brief Guide to Algebraic Number Theory by H. P. F. Swinnerton-Dyer. I like the section on page 1 'the ring of integers' as it gives a motivation for choosing which elements ...
14
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2answers
609 views

Why is $\tau(n) \equiv \sigma_{11}(n) \pmod{691}$?

If $n$ is a natural number, let $\displaystyle \sigma_{11}(n) = \sum_{d \mid n} d^{11}$. The modular form $\Delta$ is defined by $\displaystyle \Delta(q) = q \prod_{n=1}^{\infty}(1 - q^n)^{24}$. ...
14
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3answers
2k 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 ...
14
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2answers
1k views

On the ring of integers of a compositum of number fields

This is Daniel A. Marcus, Number Fields, Exercise 2.29 If anyone can help with this problem, I'd greatly appreciate it. Let $K$ be the biquadratic field $\mathbb Q[\sqrt{m}, \sqrt{n}] = \{a + ...
14
votes
1answer
783 views

What do ideles and adeles look like?

I see the ideals of an algebraic number field as lattices and prime ideals are the ones which you can't refine. How can we form a picture of ideles and adeles?
14
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1answer
243 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 ...
14
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1answer
223 views

Sum of Reciprocals of Primes in Imaginary Quadratic Field Diverges (2014 Miklós Schweitzer)

Problem 5 of the 2014 Miklós Schweitzer states: Let $\alpha$ be a non-real algebraic integer of degree two, and let $P$ be the set of irreducible elements of the ring $\mathbb{Z}[\alpha]$. Prove that ...
13
votes
5answers
703 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 ...
13
votes
2answers
603 views

Different formulations of Class Field Theory

I was reading up on class field theory, and I have a question. On wiki (http://en.wikipedia.org/wiki/Artin_reciprocity), one formulation is that there's some modulus for which ...
13
votes
2answers
763 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 ...
13
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2answers
2k views

Proof that $26$ is the one and only number between square and cube

$x^2 + 1 = z = y^3 - 1$ Why $z = 26 $ and only $26$ ? Is there an elementary proof of that ?
13
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2answers
568 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 ...
13
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1answer
176 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 ...
13
votes
2answers
359 views

Find a composite number $n$ satisfies $(2+3I)^n≡2-3I\pmod{n}$

As we know if $p$ is an odd prime number then $$(a+bI)^p\equiv a+(-1)^\frac{p-1}2bI\pmod{p},$$ where $I=\sqrt{-1}$. However, is there any composite number $n$ that satisfies ...
13
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1answer
382 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
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2answers
92 views

What is the correct analogue of $\mathbb N$ in a ring of integers?

Question: Let $K$ be a number field. The proper intuitive motivation for the ring of integers $\mathcal O_K$ is that $\mathbb Z$ is to $\mathbb Q$ as $\mathcal O_K$ is to $K$. But what plays the role ...
13
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1answer
204 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 ...
12
votes
1answer
174 views

Is $\sum\frac1{p^{1+ 1/p}}$ divergent?

Is $\displaystyle\sum\frac1{p^{1+ 1/p}}$ divergent? How can we prove that it is divergent or convergent in analytic number theory? I know what bound of the n-th prime number is, and that its order is ...
12
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1answer
404 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 ...
12
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2answers
397 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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2answers
626 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 ...
12
votes
1answer
519 views

what is a “dévissage” argument?

In Serre's "Local Fields" (Chapter 2, section 2, proposition 3) he proves something about field extensions which he breaks into parts: first he dealt with the separable case, and then with the ...
12
votes
1answer
240 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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1answer
119 views

$CL(O_S) \cong \mathbb{Z}/3\mathbb{Z}$.

Let $F = \mathbb{Q}(T)$ and let $X$ be the set of all places of $F$, and let $S = \{w\} \subset X$ where $w$ is the place of $F$ corresponding to the maximal ideal $(T^3 - 2)$ of $\mathbb{Q}[T]$. Let ...
12
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1answer
251 views

$y^2 = x^3 - 26$, exist ideal satisfying conditions?

For the solution $(x, y) = (3, 1)$ of $y^2 = x^3 - 26$, does there necessarily exist an ideal $I$ of the integer ring $\mathbb{Z}[\sqrt{-26}]$ of $\mathbb{Q}(\sqrt{-26})$ such that $(y + ...
12
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2answers
733 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
137 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
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1answer
296 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
285 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 ...
12
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0answers
68 views

Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$.

We consider an analogue of the Dirichlet $L$-function in $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, ...
12
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0answers
177 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 ...
11
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3answers
1k views

Enlightening proof that the algebraic numbers form a field

The proof I'm familiar with that the algebraic numbers $\mathbb A$ form a field uses the fact that the resultant of two polynomials $p,q\in\mathbb Q[x]$ satisfies the following properties: It is $0$ ...
11
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5answers
863 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 $(a) = aO, a \in K^*$, ...
11
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5answers
227 views

Which of these two factorizations of $5$ in $\mathcal{O}_{\mathbb{Q}(\sqrt{29})}$ is more valid?

$$5 = (-1) \left( \frac{3 - \sqrt{29}}{2} \right) \left( \frac{3 + \sqrt{29}}{2} \right)$$ or $$5 = \left( \frac{7 - \sqrt{29}}{2} \right) \left( \frac{7 + \sqrt{29}}{2} \right)?$$ ...
11
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5answers
2k 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
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 $\mathbb{Q}$. ...
11
votes
3answers
635 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 ...
11
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1answer
1k 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: ...
11
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3answers
501 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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2answers
299 views

So what *is* the Euclidean function for $\mathcal{O}_{\mathbb{Q}(\sqrt{69})}$?

It's my understanding that $\mathcal{O}_{\mathbb{Q}(\sqrt{69})}$ is UFD, PID, and Euclidean, but not norm-Euclidean. If it were norm-Euclidean, there would be a solution to $28 = q \left(\frac{5}{2} ...
11
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1answer
663 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 ...
11
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1answer
1k views

Is $\mathbb Z[\sqrt{-3}]$ Euclidean under some other norm?

I know that $\mathbb Z[\sqrt{-3}]$ is not a Euclidean domain under the usual norm $N(x + y\sqrt{-3}) = x^2 + 3y^2$, but that does not necessarily mean that it can't be a Euclidean domain. Is it ...
11
votes
1answer
1k views

How does one graduate from Hecke Operators to Hecke Correspondences?

I've read (skimmed heartily) basic books on the topic of modular forms. (The last being Silverman's Advanced Topics in the Arithmetic of Elliptic Curves.) I strive for an understanding which is as ...
11
votes
3answers
226 views

Prove that $2$, $3$, $1+ \sqrt{-5}$, and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$.

So the Norm for an element $\alpha = a + b\sqrt{-5}$ in $\mathbb{Z}[\sqrt{-5}]$ is defined as $N(\alpha) = a^2 + 5b^2$ and so i argue by contradiction assume there exists $\alpha$ such that $N(\alpha) ...
11
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2answers
160 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 ...