3
votes
1answer
36 views

Squares modulo 2^n [on hold]

How many squares are there modulo $2^n$? If we would deal with $p^n$, where p an odd prime, then we could use Hensel's Lemma, which clearly doesn't work with $2^n$.
0
votes
0answers
11 views

Can the General Number Field Sieve be used to factor in any unique factorization domain?

Related slightly to my question about factoring in quadratic rings, can you use the general number field sieve to factor in any unique factorization domain? Can you use it in any UFD that isn't the ...
1
vote
0answers
18 views

Are roots of unity in hypercomplex algebras well defined?

While playing around with cyclotomic fields, I started to wonder about taking the roots of unity in higher dimensional analogues of the complex plane. Are the roots of unity well defined in the ...
1
vote
2answers
46 views

Proof about Number Fields

It is a known result that if $\alpha$ is an algebraic integer in a number field $K$, i.e. $\alpha \in \mathcal{O}_K$, then its trace and norm are integers. I am looking over a proof of this, which ...
2
votes
0answers
20 views

Recovering congruence conditions from the Hilbert class polynomial for idoneal numbers

Before I can ask my question, I need to introduce some terminology and background. Statement 1: Let $n$ be one of Euler's 65 convenient numbers. Then we can find congruence conditions such that ...
2
votes
0answers
27 views

Does Shor's algorithm work for noncommutitive or nonassociative algebras?

Shor's algorithm is said to solve the hidden subgroup problem for finite Abelian groups, at least according to the Wikipedia article I read. This means that it can be used to solve factoring or ...
0
votes
0answers
27 views

How is the Dirichlet unit theorem effectively used to solve diophantine equations

Can someone give me an example or a link to one of its use to solve a diophantine equation ? More precisely : this theorem explains the structure of the units of the ring of integers of a number ...
4
votes
2answers
62 views

Degree of closure of $\mathbb{Q}_p$

In order to prove that algebraic closure of $\mathbb{Q}_p$ is infinite, I took the polynomial $x^n-p$ with $n>1$ over $\mathbb{Q}_p$ to show that this eqaution has no solution for infinite cases to ...
2
votes
1answer
34 views

Is the ring of integers of a local field an open subgroup?

I apologize if my question is a bit naive, but I don't have much experience in number theory and sometimes get very confused. Suppose $K$ is a non-archimedean local field (essentially a completion of ...
1
vote
1answer
54 views

Is it true that $6n^2+p$ gives primes for $n=0,1,2,\dots,p-1$ iff $Q(\sqrt{-6p})$ has class number $4$?

Let $p$ be a prime number, are the following statements true? 1.Quadratics of the form $6n^2+p$ gives primes for $n=0,1,2,\dots,p-1$ iff $Q(\sqrt{-6p})$ has class number $4$. And all such primes ...
2
votes
1answer
53 views

Existence of finite morphism to projective line

As we all know, Belyi's theorem says: A complex curve $X$ is defined over a number field, if and only if there exists a finite morphism $t:X\to \mathbb{P}^1_\mathbb{C}$ of varieties over $\mathbb{C}$ ...
2
votes
2answers
58 views

A ring automorphism in cyclotomic field

My friend asked me a question, see this. I've thought about that for some time, but I cannot do it, I don't want to let her wait too long, can you explain it for me? Thanks in advance!
2
votes
2answers
63 views

What does $conclude$ mean in this sentence?

My friend asked me a question, but I don't know the meaning of the sentence Conclude that $\sigma_n$ is a ring automorphism here, does it mean Prove that $\sigma_n$ is a ring automorphism or Make the ...
2
votes
1answer
53 views

Difference between sum of first n primes and prime(prime(n))

The seq is: -1, 0, -1, 0, -3, 0, -1, 10, 17, 20, 33, 40, 59, 90, 117, 140, 163, 218, 237, ... http://oeis.org/A239731 Is there's a formula looks like $$a(n) =n^2logn/2$$ for this seq?
3
votes
2answers
42 views

Number theory problem - contradiction

In an algebraic proof (for my problem it doesn't matter which proof) I have a special setting: $a,b,c \in \mathbb{Z}, \text{gcd}(a,c)=1,b<c \ \text{and} \ a \in \left\lbrace 1, \ldots , ...
0
votes
0answers
31 views

Zero to power Zero (Zero ^ Zero) indeterminable or not? [duplicate]

I want to know Zero power to Zero equal to 1 or Indeterminable. I think it cannot be exist. Please explain with proper mathematical definitions.
1
vote
0answers
60 views

Proposition 17, p. 68 of Lang's Algebraic Number Theory

I'm stuck on a detail of the following propositon: Let $K, E$ be linearly disjoint number fields with degrees $n$ and $m$ over $\mathbb{Q}$ whose discriminants (over $\mathbb{Q}$) are relatively ...
1
vote
0answers
23 views

Finding a particular integral basis of the cyclotomic field

Let $\zeta_{39}$ be a primitive $39$th root of unity. How can I prove that all the conjugates of $\zeta_{39}$ form an integral basis of $\mathbb{Q}(\zeta_{39})$? This is from the paper "Cyclotomic ...
3
votes
1answer
71 views

Find primitive element of splitting field of $1 + x + x^2 - x^5$

As the title says, I need to find the primitive element of the splitting field of $1 + x + x^2 - x^5$ over $\mathbb{Q}$. Firstly, I would proceed by finding the roots as the splitting field has to ...
3
votes
0answers
44 views

Find extension of $\mathbb{Q}$ containing components of eigenvectors of a matrix

Given a matrix $\mathbf{A} \in \mathbb{Z}^{d \times d}$ I need to find an algebraic number $a$ of minimal degree, such that all eigenvalues and eigenvector's coordinates of $\mathbf{A}$ belong to the ...
0
votes
4answers
74 views

Ordering the solutions to Pell's Equation

Let $S$ be the set of positive integer pairs $(x,y)$ such that $x^2 - d y^2 = -4$ or $x^2 - d y^2 = 4$, where $d$ is fixed as the discriminant of a real quadratic number field. I'm trying to show ...
2
votes
2answers
94 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 ...
5
votes
0answers
108 views

What are going to change of our view if $\pi+e$ is a rational? [closed]

It is well known that there's no conclusion now whether $\pi+e$ is a rational or not. Just for curiosity, what will happen if we know the answer?
1
vote
2answers
58 views

number theory of coefficients in an infinite sequence of polynomials

EDIT: equivalent formulation by Hurkyl in comments: if $n$ is odd and $p^\nu \parallel n$ and $n > 2k,$ then $$ p^{(\nu + 2 + 2 k - n)} \; | \; \sum_j \left( \begin{array}{c} n \\ 2j \end{array} ...
2
votes
1answer
49 views

Sum of Three squares

Considering $z=a+bi+cj$ ($a,b,c\in\mathbb{Z}$) and $w=d+ei+fj$ ($d,e,f\in\mathbb{Z}$) and the property of complex numbers that $|zw|=|z||w|$. If the rule of multiplication $zw$ is defined such that ...
1
vote
0answers
18 views

Which lattices are ideals of a number field?

Let $K$ be a number field, then its ring of integers $\mathcal{O}_K$ in the Minkowski space of $K$ is a lattice $\Lambda$. Is there some geometric descrpition/intuition that describes sublattices of ...
0
votes
2answers
49 views

Infinite primes and notation

While reading a book about algebraic number theory, the symbol for a rational prime $p$ $$p^\infty$$ often occurs and I was wondering, what the exact definition of this is. Also, what is the ...
3
votes
1answer
50 views

Cyclotomic Character

I have a couple of questions concerning the cyclotomic character. For the moment I know very little about the mod $\ell$ cyclotomic character, namely that ...
4
votes
1answer
51 views

How to determine the density of the set of completely splitting primes for a finite extension?

In reply of sea turtles comment in this thread Let $k$ be a number field and $K \mid k$ a finite Galois extension. What is the density of the set of completely splitting primes of $k$? As sea turtle ...
0
votes
1answer
43 views

Embedding into $p$-adic complex numbers

As I'm reading notes about the Leopoldt conjecture, the following question came to my mind: Let $\mathbb{C}_p$ be the $p$-adic complex numbers, i.e. the completion of the algebraic closure of the ...
0
votes
0answers
31 views

Congruences or logs

How do you know the pairs of integers $x,y$ such that $$y=\frac{ln(p(x))}{log(k)}$$ is true, where $p(x)$ is any Diophantic equation and $k$ any Natural number?
0
votes
1answer
49 views

Questions about the square root of $a$ in $\mathbb{F}_p$.

How to prove that there is a square root of $3$ in $\mathbb{F}_p$ if and only if $p \equiv 1 $ or $11 \pmod {12}$? Thank you very much.
3
votes
2answers
63 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: ...
6
votes
2answers
69 views

References for elliptic curves over schemes

As in the title, I want some references about theories for elliptic curves over rings(not fields) or over schemes. I heard that behaviours(?) of such elliptic curves are not as simple as elliptic ...
2
votes
1answer
44 views

Legendre Symbol and infinite descent method

Use infinite descent method to prove that for $p$ an odd prime: $$ \left(\frac{2}{p}\right)=-1 $$ if $$ p \equiv\pm 3\pmod 8 $$ Please I don't know how to connect two ideas. I'm sure any help ...
0
votes
0answers
79 views

Proving $\mathbb{Z}[\sqrt{-2}]$, $\mathbb{Z}[\sqrt{-1}]$, $\mathbb{Z}[\sqrt{2}]$, and $\mathbb{Z}[\sqrt{3}]$ are euclidean.

I have this short class note from my graduate number theory: ~~~~~~~~~~~~~~~~~~~~~~ THEOREM: Assume that |norm(x + y*sqrt(d))| < 1 for any two rational numbers x and y with $|x| \leq 1/2$ and ...
3
votes
1answer
79 views

solving $x^3-2y^3=1$ using cubic number field

I am trying to solve the diophantine equation $x^3-2y^3=1$ using $\mathbb{Q}(\sqrt[3]{2}).$ I've read this link: Solve $x^3 +1 = 2y^3$ The following is what i have tried: Finding all integer ...
2
votes
3answers
107 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
7answers
158 views

Trying to get a bound on the tail of the series for $\zeta(2)$

$\frac{\pi^2}{6} = \zeta(2) = \sum_{k=1}^\infty \frac{1}{k^2}$ I hope we agree. Now how do I get a grip on the tail end $\sum_{k \geq N} \frac{1}{k^2}$ which is the tail end which goes to zero? I ...
1
vote
0answers
55 views

How to solve diophantine equation $\frac{x^p-y^p}{x-y}=n$

$$\frac{x^p-y^p}{x-y}=n$$ whit $p$ a prime greater than or equal to $3$,for what value to $n$, it's solvable and how to solve,and whether $\frac{x^p-y^p}{x-y}=q_1$ $\frac{x^p-y^p}{x-y}=q_2$ is ...
1
vote
0answers
51 views

simple question regarding isomorphisms relating to ring of integers

I have a simple question about isomorphisms and ideals. Let $\mathcal O_F$ be the ring of integers in some quadratic number field $F=\mathbb{Q}(\sqrt d)$ and let $f(x)$ be the minimal polynomial of ...
3
votes
1answer
93 views

Algorithm to find solutions $(p,x,y)$ for the equation $p=x^2 + ny^2$.

As the classical book of David Cox argues, Assume the conditions are satisfied and $p$ can be represented as $x^2 + ny^2$. What would be a way to find solutions $(p,x,y)$ efficiently? Ideally, one ...
3
votes
2answers
122 views

Ring of integers for $\mathbb{Q}(\sqrt{23},\sqrt{3})$

What is the ring of integers for $\mathbb{Q}(\sqrt{23},\sqrt{3})$? So, these are numbers of the form $a+b\sqrt{3}+c\sqrt{23}+d\sqrt{69}$ where $a,b,c,d\in\mathbb{Q}$, and we want to find ones whose ...
3
votes
0answers
45 views

$P_{K,1}(\mathfrak m)\subset \operatorname {ker} \Phi_{\mathfrak m,L|K} \subset \operatorname {ker} \Phi_{\mathfrak m,M|K}$ imples $M \subset L$

Let $K$ be a number field and $L, M$ finite abelian extensions. Let $\mathfrak m$ be a modulus. Consider the two Artin maps $ \Phi_{\mathfrak m,L|K}$ and $ \Phi_{\mathfrak m,M|K}$. Let ...
6
votes
0answers
135 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 ...
2
votes
0answers
32 views

Confusing application of power residue reciprocity in Milne's CFT

Hey I am trying to figure out the details of the proof of Theorem 5.14 (p.246) in Milne's CFT (see here). I hope somebody is familiar with this. But let me sketch the proof and what I don't ...
4
votes
0answers
51 views

Prime number theorem for Dedekind domains

Let $\mathscr P\subseteq \mathbb N$ be the set of prime numbers. The prime number theorem tells us that if $\pi(x)=|\{p\in\mathscr P\colon p\leq x\}|$ then $\pi(x)\sim \frac{x}{\log x}$. Now one could ...
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 ...
0
votes
1answer
73 views

What does it mean for a “place” to divide?

A proof I'm trying to understand refers to the set of all finite places dividing an algebraic integer x. What does this mean? I can't seem to find a definition in any of the texts I've looked at. ...
2
votes
1answer
78 views

from Ireland and Rosen: when a prime remains inertial

I'm reading Ireland and Rosen's number theory book, and i'm having trouble with proposition 13.1.3 ii): Let F be $\mathbb{Q(\sqrt d)}$ where $d$ is a square free integer,and $p$ and odd prime, and ...