**1**

vote

**0**answers

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 $K$, ...

**6**

votes

**1**answer

180 views

### Property of Dirichlet character

Let $\chi_0$ be the unique Dirichlet character $\text{mod }1$ (i.e. $\chi_0(n) = 1$ for all $n$), $\zeta_p$ be a primitive $p$th root of $1$, and for any $a \in \mathbb{F}_p^\times$ and any Dirichlet ...

**9**

votes

**1**answer

256 views

### Fundamental units with norm $-1$ in real quadratic fields

If we have distinct primes $p \equiv q \equiv 1 \pmod 4,$ with Legendre $(p|q) = (q|p) = -1,$ there is a solution to $u^2 - pq v^2 = -1$ in integers and the fundamental unit of $O_{\mathbb Q(\sqrt{pq})...

**3**

votes

**1**answer

185 views

### Is the localization of the maximal abelian extension still a maximal abelian extension?

Let $K$ be a number field and consider the maximal abelian extension $K^{ab}$ of $K.$ For a finite prime $p,$ letting $K_p$ be the completion of $K$ at $p,$ we have an extension $K_p \subset K_p K^{...

**0**

votes

**0**answers

49 views

### Kernel of the Artin map when dealing with S-ideles and S-divisors for function fields

Having understood that there is a strong correspondence between number fields and function fields, I am trying to work out some function field equivalents of class field theoretic invariants from Bost ...

**7**

votes

**1**answer

209 views

### Hecke characters and Conductors

Motivation: Let $\ell$ be an odd prime. There is a conductor-preserving correspondence between primitive Dirichlet characters of order $\ell$
and cyclic, degree $\ell$ number fields $K/\mathbb{Q}$.
...

**10**

votes

**2**answers

500 views

### Equation $x^2=y^p + 1$

can you help me please for solving this diophantine equation : $x^2=y^p+1$
and if you can give me a general method to studying such equation : $x^2=y^p+t$
Thanks

**16**

votes

**1**answer

796 views

### Even Galois representations “mod p”

Consider an irreducible $\mathrm{mod}$ $p$ representation:
$$\rho: \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\bar{\mathbb{F}}_p)$$
If $\rho$ is odd, it was conjectured by Serre in ...

**4**

votes

**1**answer

204 views

### Motivation for cyclotomic units

I am wondering the original motivation for considering cyclotomic units. Maybe one can rephrase the question as:
Why did people initially consider such units in $\mathbb{Q}(\zeta_p)$ specially?
...

**24**

votes

**3**answers

830 views

### Intuition for Zagier's theorem for $\zeta_K(2)$

In 1986, Don Zagier generalized Euler's theorem ($\zeta_\mathbb{Q}(2)=\pi ^2 /6$) to an arbitrary number field $K$:
$$\zeta_K(2)=\frac{\pi^{2r+2s}}{\sqrt{|D|}}\times \sum_v c_v A(x_{v,1})...A(x_{v,s})...

**3**

votes

**1**answer

208 views

### regulators of number fields

Related question: Totally real number fields with bounded regulators
Given a number field $K$ with degree $n$ and determinant $D$, what is the "best" upper bound for its regulator $R$, if any? I know ...

**2**

votes

**2**answers

131 views

### Lubin-Tate modules and different uniformizers

Suppose I have a local field $\mathcal{O}_K$ and two different prime elements $\pi$ and $\overline{\pi},$ i.e they differ by a unit $\overline{\pi} = u \pi$ for some $u \in \mathcal{O}_K^{\times}$ not ...

**3**

votes

**1**answer

415 views

### Unramified extension of number fields

Any finite field extension (in particular Galois extension) of $\mathbb{Q}$ is ramified. Is there an intuitive geometric explanation of this fact?
Suppose we have an number field $K$, is any Galois ...

**6**

votes

**3**answers

313 views

### Argument of Zariski density to prove rationality of a regular map

Question: I want to know if the following result is correct:
Let $k$ be a number field and $k_v$ be a completion of $k$ at some place $v$, denote $K_v$ an algebraic closure of $k_v$.
Proposition.(...

**5**

votes

**1**answer

174 views

### Relation between ramification locus of a tower and of its constant field extension

I am trying to understand Remark 7.2.22 (Page 256) of Algebraic Function Fields and Codes (Second Edition) by Henning Stichtenoth.
In that remark he considers a tower $\mathcal{F} = (F_0,F_1,F_2,\...

**3**

votes

**2**answers

552 views

### What are the necessary conditions for a real number to be a cyclotomic integers？

The motivation of the question is that I try to test when a real number is not an cyclotomic integers. Or more specifically, when a positive real number is not a quantum dimension of a unitary fusion ...

**2**

votes

**0**answers

131 views

### Residual Representation of a Motive

Suppose we have $M$ a hypergeometric motive, and $\rho$ its associated Galois rep over $\mathbb{Q}_{l}$. Is there any easy/concrete way to find $\bar{\rho}$, the residual representation at a prime (in ...

**5**

votes

**1**answer

172 views

### Compact hyperbolic 3-manifolds with prescribed quaternion algebra, quaternion parameters as ramification condition

What is an interesting class of examples of hyperbolic 3-manifolds,
each of which satisfies the following conditions?
1. It is compact
2. Its trace field contains a unique imaginary quadratic ...

**4**

votes

**2**answers

384 views

### Dirichlet's approximation only using prime power as denominator

I am not sure whether this is a suitable question for MO. We know the classical version of Dirichlet's approximation theorem that if $x$ is a real number and $Q>0$ there exist $p,q\in \mathbb{Z}$ ...

**4**

votes

**1**answer

163 views

### number of generators of maximal ideals in an order of a number field

let $K$ be a number field of degree $d$ over $\mathbb{Q}$), Let $\mathcal{O}\subset K $ be an order (i.e. a $\mathbb{Z}$-lattice of $K$ contained in the integer ring $\mathcal{O}_K$ of $K$). If $ \...

**4**

votes

**0**answers

129 views

### Class field theory for $p$-groups.

I accidentally posted this question to math.stackexchange but think that it is more appropriate here (if not, please say so!):
This question is from Neukirch's book "Algebraic number theory," page ...

**3**

votes

**1**answer

131 views

### degree of Hecke field (number field of an eigenform)

Let $f\in S_k(\Gamma_1(N))$ be an eigenform, and $K_f$ be its number field, which is of finite degree over $\mathbb{Q}$. Consider the following statements.
1, $[K_f:\mathbb{Q}]=\#\{$Galois conjugates ...

**6**

votes

**1**answer

188 views

### $N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}?$

Assume that $ab \neq 0$. What is $$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}?$$I need this result, but unfortunately I am not a number theorist. Could ...

**5**

votes

**0**answers

89 views

### On a theorem of Dwork and totally ramified extensions

Suppose that $K \subset L$ is a totally abelian ramified extension of local fields. Let $\pi_L$ be a prime element of $L^*.$ $F \in Gal(\tilde{L}/L)$ is the Frobenius, where $\tilde{L}$ is the maximal ...

**20**

votes

**2**answers

991 views

### Motivating Lubin-Tate theory

The Lubin-Tate theory gives an amazingly clean and streamlined way of constructing the subfield (usually denoted) $F_\pi\subset F^\mathrm{ab}$ for a local field $F$ fixed by the Artin map associated ...

**16**

votes

**1**answer

669 views

### Are the algebraic numbers dense everywhere on the boundary of the Mandelbrot set?

Let $\mathcal{B}$ denote the boundary of the Mandelbrot set, and let
$\overline{\mathbb{Q}}$ denote the algebraic closure of the rationals.
Further put $\mathcal{B}_{\overline{\mathbb{Q}}} := \mathcal{...

**6**

votes

**2**answers

414 views

### Frobenius elements in infinite extensions

Let $K$ be a number field, $\bar K$ an algebraic closure and $G$ the associated absolute Galois group. How can I define the Frobenius elements of $G$ or at least their conjugacy class?
I know how ...

**2**

votes

**1**answer

170 views

### Ramification of prime ideal in Kummer extension

Let $\mu \in \mathbb{Q}(\zeta_n)$ lie above the rational prime $p$, and let the prime ideal $\mathscr{P}\subset \mathbb{Z}[\zeta_n]$ have ramification index $a$ over $\mu$.
Why is it then true that $...

**3**

votes

**1**answer

116 views

### Bibliography suggestion for Kummer theory

I already posted a question about a sum involving the degree of a Kummer extension.
Now I'm interested in a more specific fact about Kummer extensions.
From Hooley's paper "On Artin's conjecture", we ...

**2**

votes

**1**answer

143 views

### Doubt concerning a sum involving Kummer extension degrees

I'd like to estimate the following sum
$$
\sum_{n\leq x}\frac1{k_n}\;,\qquad x\rightarrow \infty\;,
$$
where
$k_n=[\mathbb{Q}(\zeta_n,a^{1/n}):\mathbb{Q}]$
is the degree of a Kummer extension for a ...

**32**

votes

**3**answers

979 views

### Simple argument regarding sums of two units in a number field?

I wonder if it is possible to show, without using the Schmidt subspace/Roth theorem/Baker's bounds on linear forms in logarithms or other very deep results, that, in a number field, not all integral ...

**2**

votes

**0**answers

136 views

### Maximal abelian extension and tamely ramify extension

Let $K$ be a number field and $v$ a finite place of $K$ lying above a prime number $p$. Assume further that $v$ is unramified over $p$. If $K^{ab}$ is the maximal abelian extension let $w$ be a ...

**4**

votes

**1**answer

279 views

### Completion of a finite field extension is also finite?

Let $(L,w)/(K,v)$ be a finite extension of valuation fields, and let $L_w$, $K_v$ be the respective completions of $(L,w)$, $(K,v)$. Is the field extension $L_w/K_v$ finite?
For nonarchimedean ...

**8**

votes

**0**answers

190 views

### Geometric meaning of conductor

Supppose $L/K$ is a finite extension, choose $\theta \in O_L$ such that $L=K(\theta)$. We define the conductor of ring $O_K[\theta]$ to be an ideal of $O_L$, namely: $F=\{\alpha\in O_L|\alpha\cdot O_L\...

**2**

votes

**0**answers

135 views

### calculation in a group ring

I have some problems with the verification of the third equation in Lemma 1 in this paper.
First of all, one has to notice that there is at least one Error in the Definition of $a_{\kappa,\nu}$ above....

**9**

votes

**1**answer

608 views

### Elementary proof of a special case of Chebotarev's density theorem

A special case of Cheboratev's density theorem states that, for $K/\mathbb{Q}$ a Galois number field of degree $n$, then the rational primes that split completely in $K$ have density $1/n$.
Is there ...

**5**

votes

**0**answers

101 views

### On existence of rapid Arithmetic geometric procedure?

We know that $\pi$ can be computed by Arithmetic Geometric mean using Gauss-Legendre procedure which does provide fastest convergence rate as well with a guarantee of $2^n$ bits of $\pi$ at $n$th ...

**15**

votes

**1**answer

826 views

### A set of generators for $\bar{\mathbb{Q}}$

Two questions:
Does there exist a sequence $\alpha_1,\alpha_2,...$ of algebraic numbers with degrees $d_1,d_2,...$ s.t. for each $i$, $d_i|d_{i+1}$ and $\alpha_i= p_i(\alpha_{i+1})$ with $p_i$ a ...

**10**

votes

**0**answers

209 views

### What is the relationship between the conductor of an order and the conductor of a number field extension?

What is the relationship between the conductor $\mathfrak{f}_{\mathcal{o}}$ of an order $\mathcal{o}\subset \mathcal{O}_K$ and the conductor $\mathfrak{f}_{L/K}$ of a field extension in the classical ...

**7**

votes

**2**answers

520 views

### On bounds for idoneal integer

What is the best known lower bound and upper bound known for such a number if it exists and have there been any attempts (computational including) to eliminate the existence of such a number in known ...

**9**

votes

**0**answers

174 views

### The operator $\left(q\frac{d}{dq}\right)^s$ and fractional derivatives of modular forms

Recall the notion of a "nearly holomorphic modular form" introduced by Shimura:
A function $f : \mathfrak h \to \mathbb C$ is said to be nearly
holomorphic of level $\Gamma_1(N)$, weight $k$ and ...

**1**

vote

**2**answers

173 views

### Counting number of $2\times 2$ unimodular matrices of particular type

From set of numbers from $\Bbb S=\{0,1,\dots,m\}$, how many distinct $3\times 3$ unimodular matrices parametrized by $(a,b,c,d,e,f)\in\Bbb S^6$ of following type can one form?
\begin{bmatrix}
a^2 &...

**1**

vote

**0**answers

75 views

### Statements generalizing representability of integers by binary quadratic forms to $n$-variable higher homogeneous forms?

Representing integers through the theory of binary quadratic forms is a well studied topic. We know that given $a,b,c\in\Bbb N$, based on discrimant $b^2-4ac$, we can study the representability of ...

**0**

votes

**0**answers

158 views

### Writing integers in ring of integers of number fields

Given $a,b\in\Bbb N$, we can write $a=a_tb^t+a_{t-1}b^{t-1}+\dots+a_1b+a_0$ where $t=\lceil\log_ba\rceil$ and $a_i<b<a$.
(1) Supposing if $b\in\mathcal{O}_K$ where $\mathcal{O}_K$ is ring of ...

**2**

votes

**0**answers

154 views

### For $K/E$ a number fields extension and $F/E$ a finite Galois extension, how is the ramification in $F\cdot K/K$ related to the one in $F/E$?

Studying class field theory, I have come across the following Proposition:
Proposition. Let $K/E$ be an extension of number fields so that there is no nontrivial unramified subextension $F/E$ with $...

**8**

votes

**2**answers

557 views

### Does the Galois group of a Pisot polynomial contain the alternating group?

Let $n \in \mathbb{N}$, and let $p(X) \in \mathbb{Z}[X]$ be a monic polynomial of degree $n$. Suppose that exactly one complex root of $p$ is of modulus $> 1$, and that the remaining $n-1$ roots of ...

**2**

votes

**2**answers

217 views

### Decomposition and valuation rings

I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theorem:
$(K,u)$ be a field and its absolute value, $(K_u,\bar u)$ be its completion and absolute value ...

**6**

votes

**1**answer

186 views

### Can a product of conjugates be a Pisot number again?

Let $p(X) \in \mathbb{Z}[X]$ be an irreducible polynomial, and let $\alpha_1 \dots, \alpha_n$ be its roots in $\mathbb{C}$. Suppose that $\alpha_1$ is a Pisot number, that is, $\alpha_1 \in \mathbb{R},...

**2**

votes

**1**answer

195 views

### On Pell's equation

A post was made (Reduction from factoring to solving Pell equation) seeking clarification to solving $$x^2-Dy^2=1$$ to factoring when $D>0$. An answer was posted stating that to factor $N$, it ...

**2**

votes

**1**answer

144 views

### Unique extension of the absolute value

Let $(K,u)$ be a complete valued field, $u$ be its discrete absolute value (corresponds to a discrete valuation on $K$), then:
($\ast)$ Let $E/K$ is a finite separable field extension, then the ...