**-2**

votes

**0**answers

19 views

### Function such that f(a,b) = c, but even if I knew c and b I cannot (practically) find a? [closed]

I need a function where f(a,b) = c. But even if you knew b and c you cannot practically discover a. I.e. not Cantor Pairing because in Cantor Pairing both a and b can easily be discovered if just c ...

**11**

votes

**2**answers

645 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 ...

**3**

votes

**0**answers

115 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

**1**answer

207 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

**0**answers

107 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

**1**answer

237 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

**0**answers

159 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

**1**

vote

**1**answer

162 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

**1**answer

376 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

**1**answer

129 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 $L/\mathbb{Q}$ ...

**7**

votes

**0**answers

111 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) = ...

**15**

votes

**1**answer

640 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

**1**answer

166 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 ...

**19**

votes

**1**answer

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

**1**answer

108 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

**0**answers

120 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

**0**answers

123 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 ...

**2**

votes

**1**answer

270 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

**1**answer

726 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

**1**answer

257 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

**0**answers

45 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 $K$, ...

**5**

votes

**0**answers

94 views

### Property of Dirichlet character

Let $m$ be an integer prime to $p$ such that $\chi^m = \chi_0$ on elements of $\mathbb{F}_p^\times$. We let $\zeta_m$ be a primitive $m$th root of unity. For $b$ any integer prime to $m$ define ...

**9**

votes

**1**answer

212 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 ...

**3**

votes

**1**answer

155 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 ...

**5**

votes

**1**answer

171 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

489 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

**11**

votes

**1**answer

531 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

194 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

756 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 ...

**3**

votes

**1**answer

172 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

117 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

354 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

299 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 $k_v$.
...

**2**

votes

**1**answer

121 views

### How many points are in such set with the same norm-2

Let $L=[a,b]\cap\mathbb{N}$ with $a,b\in\mathbb{N}$, let $D\in\mathbb{N}$, and let $C=L^D$. Then I would like to know how many points are there in $C$ with the same given norm-2 $d$. I.e., I'm looking ...

**3**

votes

**2**answers

535 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

106 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

139 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

352 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

150 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

113 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

112 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

186 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 ...

**4**

votes

**0**answers

79 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 ...

**17**

votes

**2**answers

698 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

649 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}}} := ...

**6**

votes

**2**answers

374 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

135 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

109 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

139 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 ...

**30**

votes

**3**answers

949 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 ...