Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale ...

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2
votes
1answer
132 views

Criteria for the surjectivity of the reduction map of the $SL_n$-group scheme

Let $R$ be a commutative ring and let $I\subseteq R$ be an ideal. We have a natural projection map $$ \pi:SL_n(R)\rightarrow SL_n(R/I) $$ (In the original question I had put $GL_n$ instead of $SL_n$ ...
1
vote
0answers
96 views

Combinatorial splitting in number rings

The goal of this problem is to see if there is a structured way to factor numbers constructed from a set of distinct odd primes $p_1$ through $p_n$ in a number ring. Take an arbitrary non empty ...
11
votes
1answer
201 views

A Galois extension over $\mathbb{Q}$ with Galois group $A_4$ and with cyclic decomposition groups

Does there exist a Galois extension $L/\mathbb{Q}$ with Galois group $A_4$ (the alternating group on four letters) such that all the decomposition groups are cyclic? This question is motivated by the ...
0
votes
0answers
55 views

Congruent numbers and primorials

The first 10 primorials (2, 6, ... , 6469693230) are congruent numbers subject to the Birch Swinnerton-Dyer conjecture. My question is - What is the first primorial not to be a congruent number (...
0
votes
0answers
34 views

Is that possible to use stieltjes transform for multiple matrices

I have the matrix calculation with expression \begin{equation} \frac{1}{M}tr(\mathbf{WHH}^H\mathbf{W}^H + \mathbf{R}_{nn})^{-1} \end{equation} whereas $\mathbf{H} \in \mathbb{C}^{M\times K}$, $\mathbf{...
7
votes
0answers
263 views

Capitulation of ideal classes in general Dedekind Domains

I’ve been working on a problem, and come across an issue with capitulation in Dedekind domains. Here is the set up: Let $D$ be a Dedekind domain, and $K$ its (perfect, but we’re willing to modify ...
2
votes
1answer
221 views

Non-negative integer solutions of x^2+y^3=n

I have the next equation: $x^2+y^3=n$. Where n is a positive integer constant. I want to know the exact number of non-negative integer solutions. Also I want to know what are those solutions. How ...
8
votes
1answer
222 views

Imaginary quadratic fields: Euclidean if and only if norm Euclidean

Let $K$ be an imaginary quadratic field and $O_K$ be its ring of integers. We say $O_K$ is norm Euclidean if the norm is a Euclidean function. It is known from the classification of imaginary ...
21
votes
4answers
1k views

Hasse principle for rational times square

Does a Hasse principle hold for the property of being a rational times a square ? Let $a \in \mathbb{K}$ be an element of a number field. Assume that at every place $\mathbb{K}_v$ of $\mathbb{K}$, $a$...
2
votes
0answers
54 views

Prescribed norm residue symbol in number field

Suppose $F$ is a number field, and $a, b$ are non-zero elements. Does there always exist $x \in F$ such that the norm residue symbols (=cup products) are $(a, x)= 0 = (x, b) \in H^2(F, \mathbb{F}_2)$ ...
0
votes
1answer
220 views

On the quadratic reciprocity law? [closed]

In the Quadratic Reciprocity Law $$\exists x\in\Bbb{N}\quad x^2\equiv p\pmod q\iff\exists y\in\Bbb{N}\quad y^2\equiv q\pmod p$$ if $p\equiv q\equiv 1\pmod4$. Is there any relation between $x$ and $y$ ...
2
votes
3answers
381 views

Learning roadmap for algebraic number theory

I have read some elementary number theory from David Burton's text and I know groups and rings from Herstein's book Topics in Algebra and some field theory from different sources online. I am ...
5
votes
0answers
87 views

What is the precise relationship between primitive Hida families and the connected components of the ordinary locus of the eigencurve?

In the references I've found discussing this question, I have not found any statements that I can understand and that are as precise as I would like. I'm more familiar with Hida families than with the ...
13
votes
3answers
454 views

Infiniteness of the Galois cohomology over a number field with coefficients in a finite Galois module

Let $k$ be a number field and $M$ be a nonzero finite discrete $\mathrm{Gal}(\bar k/k)$-module. Is it true that $H^1(k,M)$ is infinite? This would complete the answer of Daniel Loughran. There is a ...
5
votes
0answers
111 views

Primitive element for a number field, and ramification

Let $K=\mathbb Q(\theta)$ be a number field with integral primitive element $\theta$, and let $f(x)$ be the minimal polynomial of $\theta$. Let $p$ be a rational prime. It's well known that if $p$ ...
3
votes
0answers
101 views

Globalizing local field extensions with controlled ramification

Let $K_1/k_1, \ldots, K_r/k_r$ be "separable cyclic" extensions of degree $n$ where each $k_i$ is a local field of characteristic 0 (archimedean or not). By separable cyclic of degree $n$, I mean the ...
1
vote
0answers
74 views

Exceptional primes in Kummer-Dedekind theorem

Suppose that $A$ is a Dedekind domain with fraction field $K$, $L$ is a finite separable extension of $K$, and $B$ is the integral closure of $A$ in $L$. Suppose that $t$ is a primitive element for $L/...
1
vote
0answers
71 views

“Algebrazing” canonical subgroups of elliptic curves

I'm puzzled by a part of the construction of the canonical subgroup of a "not too supersingular" elliptic curve. In Katz's paper, one produces a subgroup of the formal group of the elliptic curve but ...
1
vote
2answers
311 views

Expository articles on Algebraic Number Theory

I am about to start learning Algebraic Number Theory and thus was looking for some expository articles on this subject. So far I have found two such articles: Dickson, L. E.. (1917). Fermat's Last ...
2
votes
1answer
163 views

Fundamental Units in Totally Real Cubic Fields

How much is known about the fundamental units in totally real cubic fields? For example, Daniel Shanks has a family of totally real cubic fields for which the fundamental units are known; those with ...
2
votes
1answer
148 views

Character group of the multiplicative rationals

I was reading some stuff on Hecke characters and came across an issue I have not been able to resolve. I posted it here on math stack exchange first. Let $\mathbb{Q}^{\times}$ be the multiplicative ...
3
votes
1answer
185 views

Connected-étale sequence for ordinary CM elliptic curves

Let $E/k$ be an elliptic curve over algebraically closed field of characteristic $p$ with CM, for simplicity, by the maximal order of a quadratic imaginary field $K/\mathbb{Q}$. Suppose that $p$ is ...
10
votes
1answer
235 views

On $\eta(6z)\eta(18z)$ and the splitting / modularity of $x^3 - 2$

Consider one of the simplest non-abelian examples of modularity. Let $$\eta(6z)\eta(18z) = q\prod_{n=1}^\infty (1 - q^{6n})(1 - q^{18n}) = q - q^7 - q^{13} -q^{19} + q^{25} + 2q^{31} - q^{37} + 2q^{43}...
2
votes
3answers
343 views

Primes in arithmetic progressions in number fields

My general question is how does one prove equi-distribution results for primes in arithmetic progressions in number fields? I am interested in the equi-distribution of prime elements of the ring of ...
9
votes
2answers
543 views

Number of polynomials whose Galois group is a subgroup of the alternating group

Let $f = x^n + a_{n-1}x^n + \cdots + a_0$ be a monic polynomial of degree $n \geq 2$ with integer coefficients. By $\text{Gal}(f)$ we mean the Galois group over $\mathbb{Q}$ of the Galois closure of $...
1
vote
0answers
79 views

Why is the kernel of an algebraic Hecke character open in the ideles?

I've been reading about algebraic Hecke characters, and how one obtains one dimensional $p$-adic representations from them. I have a question about why the kernel of a map defined on the ideles is ...
23
votes
0answers
262 views

Does every ring of integers sit inside a monogenic ring of integers?

Given a number field $K/\mathbf{Q}$ whose ring of integers $\mathcal{O}_K$ is, in general, not of the form $\mathbf{Z}[\alpha]$ (not monogenic), does there exist an extension $L/K$ which has $\mathcal{...
1
vote
0answers
147 views

What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$? [closed]

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$. If $L$ is a number field ...
0
votes
0answers
105 views

More generalized RSA construction

Is there a way to construct RSA type cryptosystem over general number rings? Can Number Field Sieve technique be applied here?
4
votes
0answers
63 views

An order in $\mathbb Q[G]$ which is a maximal $\mathbb Z_p$-order in $\mathbb Q_p[G]$ for finitely many primes $p$

Let $G$ be a finite group and $S$ a finite set of prime numbers. I know that every separable $\mathbb Q$-algebra $A$ contains a maximal $\mathbb Z$-order but I wonder if the following is true. Is ...
4
votes
2answers
275 views

The best possible density in Hilbert's Irreducibility Theorem

Let $f(X,t_1,\dots,t_s)$ be an irreducible polynomial with coefficients in $\mathcal{O}_K$, the ring of integers of a number field $K$. By work of S. D. Cohen (http://plms.oxfordjournals.org/content/...
7
votes
1answer
209 views

If two Hecke characters cut out the same field, are they Galois conjugates?

First question on MathOverflow, I hope it is appropriate for this site. There are two related questions. Let $K$ be a number field, $G_K = Gal(\overline{K}/K)$, $p$ a prime, and $$\chi_1,\chi_2:G_K\...
5
votes
2answers
196 views

Mahler measure of a totally positive, expanding algebraic integer

Consider a degree-$d$ algebraic integer $\alpha$ all of whose conjugates (including itself) are real numbers greater than 1. Its Mahler measure $M(\alpha)$ is simply equal to the norm $N(\alpha)$. ...
3
votes
0answers
114 views

How to Taylor series expand at the prime at infinity

Given a rational number, one can find a Taylor series expansion with respect to any $p$-adic valuation, as covered in Gouvea's introductory text on $p$-adic numbers. My question is how does one do ...
9
votes
0answers
461 views

Algebraic proofs of algebraic theorems about algebraically closed fields

It is well-known that the first order theory of algebraically closed fields admits quantifier elimination, whence the theory $ACF_p$ of algebraically closed fields of given characteristic $p$ is ...
12
votes
1answer
214 views

Are quadratic units cyclotomic norms?

Consider the fundamental unit $\varepsilon$ of a real quadratic number field $k = {\mathbb Q}(\sqrt{p})$ for primes $p \equiv 1 \bmod 4$, and let $h$ denote its class number. By Dirichlet's work on ...
3
votes
0answers
117 views

How to show that $h(-D)\geq \displaystyle\sum_{a\in A_1\\, 1\leq a\leq{\frac{\sqrt D}{2}}} 1$?

Here $A_1=\{u;p|u\Longrightarrow \chi(p)=1\}$ with $\chi$ a real quadratic character and $h(-D)$ the class number of the imaginary quadratic field of the fundamental discriminant. This problem occurs ...
11
votes
3answers
571 views

Philosophy behind cohomological representations

For a given real reductive Lie group $G$, we have the notion of a representation being cohomological using the Lie algebra cohomology. In particular we know that the discrete series representations of ...
11
votes
5answers
1k views

How much do I need to learn algebraic geometry to understand arithmetics over number fields

I am at the stage of learning. Mostly, I am attracted by algebraic number theory. Roughly speaking, I am interested in the rational points of algebraic varieties. I am little bit afraid to start to ...
16
votes
2answers
742 views

Special topics to include in course in algebraic number theory

I'll be teaching an introductory course in algebraic number theory this fall (stopping before class field theory). I'm looking for a good list of "special topics" I can include to illustrate the ...
-1
votes
1answer
69 views

Equation with norms of cyclic extensions of coprime degrees

Let $\mathbb{K}$ be a quadratic extension of $\mathbb{Q}$ and $\mathbb{L}$ be a cyclic extension of $\mathbb{Q}$ of odd degree. Given a rational $r\neq 0$, does there always exist $k\in \mathbb{K}^*$ ...
3
votes
0answers
105 views

Siegel's article “The volume of the fundamental domain for some infinite groups”: trouble with understanding computations

This is the article I mentioned. While the idea of what Siegel is doing in order to compute the volume of the fundamental domain described in the article (the very first one, for there are discussed ...
4
votes
0answers
151 views

Computing algebraic properties of trace fields, as given by SnapPy

SnapPy can tell you the trace field of a hyperbolic $3$-manifold (which is awesome), but it specifies the field by outputting: the minimal polynomial of the field over $\mathbb{Q}$, and a decimal ...
12
votes
1answer
214 views

A prime ideal $\mathfrak{p}$ decomposes in $\mathbb{Q}(\zeta_{24})/\mathbb{Q}(\sqrt{-6})$ iff it is generated by $\alpha\in1+2\Bbb{Z}[\sqrt{-6}]$

For a nonzero prime ideal $\mathfrak{p}$ of $\mathbb{Z}[\sqrt{-6}]$ which does not divide $2$, does $\mathfrak{p}$ decompose completely in the extension $\mathbb{Q}(\zeta_{24})/\mathbb{Q}(\sqrt{-6})$ ...
24
votes
1answer
529 views

Artin reciprocity $\implies $ Cubic reciprocity

I asked this on math.SE a few days ago with no reply, so I'm reposting it here. Hope this is not considered too elementary for MO (feel free to close if so). I'm trying to understand the proof of ...
3
votes
0answers
75 views

Sign of bivariate polynomial evaluated over two algebraic numbers

I would like to compute the sign of a bivariate polynomial $f$ evaluated over two algebraic numbers $a$, $b$. The numbers are in "isolating interval representation" meaning that each one is defined by ...
4
votes
0answers
142 views

extending $p$-adic character of the local intertia to the absolute Galois group

Suppose I have a number field $F$, and a finite place $v$ of $F$. Let $E$ be finite extension of $F_v$. I start with a continuous morphism $$ \chi \colon O_{F_v}^\times \to E^\times. $$ where $O_{F_v}$...
6
votes
3answers
261 views

For an arithmetic hyperbolic 3-manifold group, when is its trace field not its invariant trace field?

Edit: In my original post I failed to require the group to be a manifold group. The answer below from @BenLinowitz works in that case. I am really interested though in when the group is torsion-free, ...
5
votes
0answers
108 views

Factorization problem in Cyclic cubic field

Let K/$\mathbb{Q}$ be a cubic number field. Assume that K/Q be Galois with class number 1. Therefore Gal(K/Q) is cyclic cubic group and $\mathcal{O}_K$ is a PID. Let p be a rational prime, p ...
6
votes
2answers
341 views

Are the abelian absolute Galois groups of these local fields isomorphic?

For a field $F$ we denote by $F^{\mathrm{ab}}$ the compositum of all finite Galois abelian extensions of $F$. Is $\mathrm{Gal}(\mathbb{Q}_2(\sqrt[8]{3})^{\mathrm{ab}}/\mathbb{Q}_2(\sqrt[8]{3})) \...