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 ...

learn more… | top users | synonyms

9
votes
0answers
471 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
227 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
581 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
746 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
70 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
107 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 ...
6
votes
1answer
206 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
216 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
590 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
154 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
266 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
112 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
350 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})) \...
18
votes
1answer
719 views

Concrete Applications of knowing $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$

I have very little experience with Galois representations, mostly as they relate to class field theory, elliptic curves, and modular forms, but they seem to have quite a reputation in number theory as ...
10
votes
1answer
386 views

What was a cusp to Hurwitz in 1892?

Let $d\in\mathbb{N}$ be squarefree. Let $\mathcal{O}_d$ be the ring of integers of $\mathbb{Q}(\sqrt{-d})$. Let $\Gamma_d=\mathrm{PSL}_2(\mathcal{O}_d)$. Let $\mathcal{H}^3$ be the upper half-space ...
0
votes
1answer
54 views

Norm Residue Symbol refinement?

From Wikipedia: given $a\in K^\times$, (a,b)=1 for all b [in K*] if and only if a is in K*ⁿ So suppose that $(\frac{a\ ,\ K^\times\!}{p})\neq 1$ [assume $n$ above ...
12
votes
3answers
700 views

Profinite groups as absolute Galois groups

It is a well-known result that all profinite groups arise as the Galois group of some field extension. What profinite groups are the absolute Galois group $\mathrm{Gal}(\overline{K}|K)$ of some ...
14
votes
1answer
479 views

Growth of $\zeta_{\mathbf Q[\cos(\frac{\pi}{2^{n+1}})]}(2)$

Let $K_n$ be the field $\mathbf Q[\cos(\frac{\pi}{2^{n+1}})]$ (the real subfield of the cyclotomic field $\mathbf Q[e^{\frac{i\pi}{2^{n+1}}}]$). Is there anything known about the growth of the ...
9
votes
1answer
303 views

Parametrizing all cyclic extensions of the rational numbers of degree 5

Is there a polynomial $f(T,X) \in \mathbb{Q}(T)[X]$ in the indeterminate $X$ over the field $\mathbb{Q}(T)$ with $\mathrm{Gal}(f/\mathbb{Q}(T)) \cong \mathbb{Z}/5\mathbb{Z}$ such that for every Galois ...
15
votes
1answer
373 views

Is the number of representations as the sum of two elements of a polynomial sequence always small?

Let $f(x) \in \mathbb{Z}[x]$ be a degree $d>1$ polynomial with integer coefficients. Define $$r(n) := | \{x,y \in \mathbb{Z} : f(x)+f(y) = n \}|. $$ My question is: Is it true that $r(...
1
vote
1answer
120 views

Existence of class modules for finite groups

I asked the following question on Stackexchange and got no reply so I am reposting it here. Let $G$ be a finite group. A $G$-module C is a class module if, for all subgroups $H \subset G$: 1) $H^1(H,...
16
votes
2answers
858 views

Can the Dedekind zeta function distinguish between real and imaginary quadratic number fields?

Suppose I am given a machine that gives me the coefficients $a_1$, $a_2$, $a_3$, ... of a Dirichlet series $$\sum_1^{\infty} \frac{a_n}{n^s} $$ and assume that I know that this Dirichlet series is the ...
35
votes
13answers
3k views

Applications of the Cayley-Hamilton theorem

The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own ...
4
votes
1answer
181 views

What can we say about the differences between roots of a polynomial with large Galois group?

Suppose that $K$ is a number field and $L$ is the splitting field of a monic polynomial in $\mathcal{O}_{K}[x]$ of degree $d \geq 5$ with roots $\alpha_{1}, ... , \alpha_{d}$. Assume that the $\...
5
votes
0answers
367 views

Unique quadratic subextension of a ray class field

Let $K_q$ denote the unique quadratic subextension of the ray class field over $\mathbb{Q}$ of conductor $q\times\infty$. Then $K_q$ should be $\mathbb{Q}(\sqrt{q})$ if $q$ if 1 mod 4 and $\mathbb{Q}(\...
1
vote
0answers
112 views

Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers upto $\frac{1}{2}p^2_x -2$ exvert all non-trivial $\Xi_x$-classes? [closed]

This question follows from the information provided below. Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers up to $\frac{1}{2}p^2_x -2$ exvert all non-trivial $\...
22
votes
2answers
801 views

Elementary congruences and L-functions

In a recent article, Emmanuel Lecouturier proves a generalization of the following surprising result: for a Mersenne prime $N = 2^p - 1 \ge 31$, the element $$ S = \prod_{k=1}^{\frac{N-1}2} k^k $$ ...
12
votes
2answers
721 views

Upper bound on answer for Pell equation

A user on MSE, @martin , asked http://math.stackexchange.com/questions/1611411/pell-equations-upper-bound about an upper bound for $x$ in $x^2 - p y^2 = 1,$ when $p$ is prime. I checked, it appears ...
5
votes
0answers
162 views

Iwasawa theory, $\mathbb{Z}_p^{2}$-extension, Greenberg module

Take $H\subset \bar{\mathbb{Q}}$ be a quartic imaginary number field such that $\operatorname{Gal}(H/\mathbb{Q})=\mathbb{Z}_2 \times \mathbb{Z}_2$. Denote by $F$ the quadratic real subfield of $H$ and ...
2
votes
1answer
227 views

Question related to Fermat curve: Does the equation $A x^n + By^n = C z^n$ have any solution in $\mathbb{N}$?

Let $A, B, C \in \mathbb{N}$ be such that $\gcd(A,B,C)=1$. Is it known if the equation $A x^n + By^n = C z^n$ has any non-trivial solutions $x,y,z \in \mathbb{N}$? I know there are no such solutions ...
4
votes
0answers
110 views

Characteristic zero lifts of a mod 4 cusp form

Let $E$ be an non-CM elliptic curve over $\mathbb{Q}$ of conductor $N$ with a cyclic rational $4$-isogeny and let $f$ denote the corresponding cuspidal eigenform of level $N$. Let $E_4$ denote the ...
3
votes
1answer
250 views

When is possible to factor a field extension into one which adds no roots of unity, followed by one which adds only roots of unity?

The answer to whether this is possible for general fields is no. However, the counterexamples used two ingredients: 1) $\Bbb Q_p$, whose extensions $K$ containing $\Bbb Q_p(\sqrt[p^e]{u})$ might not ...
0
votes
0answers
186 views

Why is Kronecker's Jugendtraum only for Abelian extensions?

Why is Kronecker's Jugendtraum only for Abelian and not for more general extensions of number fields? Wikipedia, Hilbert's Twelfth Problem
2
votes
1answer
183 views

Generalizing Dedekind's Factorization Theorem

A classical theorem due to Dedekind states the following: Let $O_{K}$ be the ring of integers of a number field $K$, and assume $K$ is generated by adjoining the algebraic integer $\alpha$ to ...
10
votes
1answer
402 views

What are “Artin fractions”?

The German Wikipedia entry for Ernst Witt https://de.wikipedia.org/wiki/Ernst_Witt has a photo of his grave in Hamburg. The bottom part has a visible text "Artin Brueche" (Artin fractions) but the ...
6
votes
1answer
167 views

Unramified extensions of quadratic fields

Let $K/\mathbb{Q}$ be quadratic and let $L/K$ be an (everywhere) unramified Galois extension. If $L/K$ is abelian, then one can show that $L/\mathbb{Q}$ is Galois (eg see here). Is $L/\mathbb{Q}$ ...
8
votes
0answers
125 views

Equation which has nontrivial solutions modulo $N$ for every $N \ge 2$ does not have any nontrivial integer solutions

Let $\alpha = \sqrt[3]{2}$ and $K = \textbf{Q}(\alpha)$. I want to show that the equation$$\text{N}_\textbf{Q}^K\left(x + 4y + z\alpha + w\alpha^2\right) - 6(x + y)\left(x^2 + xy + 7y^2\right) = 0,$$...
15
votes
1answer
667 views

What's so special about these $17$th deg equations?

While browsing the Database of Number Fields, I came across 17T8. It only had four equations, one of which is, $$\small{x^{17} - 5x^{16} + 40x^{15} - 140x^{14} + 610x^{13} - 1622x^{12} + 4870x^{11} - ...
5
votes
1answer
179 views

Applications of the Galois embedding problem

Given a finite Galois extension of number fields $L/K$ with Galois group $G$ and a surjection $E\twoheadrightarrow G$ of finite groups, the Galois embedding problem is the question of whether there ...
20
votes
1answer
1k views

What's special about the circle problem?

Let $K$ be a number field, and let $$\zeta_{K}(s):= \sum_{0 \neq I \text{ ideal of }O_K} \frac{1}{N_{K/\mathbb{Q}}(I)^s} = \sum_{n \ge 1} \frac{a_n}{n^s}$$ be the Dedekind zeta function of $K$. The ...
4
votes
1answer
118 views

Exceptional specializations of Galois groups in the Hilbert Irreducibility Theorem

Suppose $f(x,t)\in\mathbb{Q}(t)[x]$ is an irreducible polynomial with Galois group G. For any rational number $a$ we may consider the polynomial $f(x,a)\in\mathbb Q[x]$ and its corresponding Galois ...
9
votes
0answers
136 views

Newly defined $L$-function in terms of $L$-function, does it have any obvious zeros or poles?

Let $K$ be a number field, $Cl(K)$ the ideal class group, $\chi: Cl(K) \to \mathbb{C}^\times$ a homomorphism. If $\mathfrak{a} \subset \mathcal{O}_K$ is any ideal, let $[\mathfrak{a}]$ denote its ...
5
votes
0answers
130 views

Particular case of the class number formula, Dirichlet characters

Let $\chi$ be a Dirichlet character modulo $4$ such that $\chi(-1) = -1$, and let $\chi'$ be a Dirichlet character modulo $5$ such that $\chi'(-1) = 1$, $\chi'(2) = \chi'(3) = -1$. How do I see the ...
5
votes
1answer
524 views

On the Diophantine equation $x^2 = y^p + 2^{r}z^p$ where $p\geq 7$ is an odd prime and $r \geq 2$

It is known that the only nonzero pairwise coprime integer solutions to the above Diophantine equation are for $r=3$, for which $(x, y, z) = (3,1,1)$ and $(-3,1, 1)$. (Cohen, Number Theory Volume 2: ...
12
votes
1answer
762 views

On cubic reciprocity for $x^3+y^3+z^3 = 996$?

I. The Diophantine equation, $$x^3+y^3+z^3 = 3w^3\tag1$$ with $x\geq y \geq z$ and $w=1$ has only two known solutions, namely $1,1,1$ and $4,4,-5$. Are there larger ones? As Noam Elkies points out ...
1
vote
1answer
271 views

How can one show that the hyperelliptic curve $y^2 = x^{p} + \frac{1}{4}$ has only one positive rational solution for every prime $p>3$?

Without applying Fermat's Last Theorem, how can one show that the hyperelliptic curve $y^2 = x^{p} + \frac{1}{4}$ has only one positive rational solution $(x,y) = (0, \frac{1}{2})$ for ever prime $p \...
1
vote
0answers
47 views

Normgroup and the image of the Hilbert symbol are subgroups of index 2 in the principal units

Let $K$ be a local field over $\mathbb{Q}_2$ such that the extension $K(i)/K$ is ramified and let $U^1_{K(i)}$ and $U^1_K$ denote the groups of principal units in the fields $K(i)$ and $K$, ...