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### Number of solutions of the congruence, $x-y \equiv z \pmod{n}$,$x,y\in U_{n}$?

I known number of solution of the congruence,
$x+y \equiv z \pmod{n}$,$x,y\in U_{n}$ is $N(z)=n\prod_{ p\backslash n}\left(1-\frac{\varepsilon(p)}{p}\right)$,where $\varepsilon(p)$=$\left\{
...

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222 views

### Formal group law over $\mathbb{F}_p$

Let $p$ be a prime. For each $n > 0$ there is a unique 1-dimensional commutative formal group law $F$ over $\mathbf{Z}$, $F(X, Y) = X + Y + \dots \in \mathbf{Z}[[X, Y]]$, whose logarithm function ...

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**1**answer

612 views

### What are the products $\prod_{A\subset{\mathbb F}_p\colon |A|=n} \sum_{a\in A} \zeta^a$ equal to?

This is a somewhat more explicit version of a question I have recently asked.
Let $p$ be an odd prime, and write $\zeta:=\exp(2\pi i/p)$ (any other primitive $p$th root of unity will do as well). For ...

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**1**answer

103 views

### Fixed field of the Nebentypus of a newform for $\Gamma_1(N)$

Let $f=\sum_{n\geq 1}\in S_2(\Gamma_1(N),\varepsilon)$ be a normalized newform without CM and with Nebentypus $\varepsilon$. Let $L=\mathbb Q(a_n\colon n\in \mathbb N)$ be the number field generated ...

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**1**answer

394 views

### An old paper of S.Chowla on unit equations

It is referenced that in
Chowla, S., Proof of a conjecture of Julia Robinson, Norske Vid. Selsk. Forh. (Trondheim) 34, 100–101 (1961),
it is shown that the equation $\epsilon_1 + \epsilon_2 = 1$ ...

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311 views

### Quintic polynomials generating cyclic extensions

We know that a cubic equation generates a cubic cyclic extension iff it has a perfect square discriminant. Now I am wondering if there is a similar condition for quintic polynomials. So I am trying to ...

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87 views

### Prime factors of $\sum_{i\in I} \zeta_p^i$

For a rational prime $p>3$, denote by $\zeta$ a fixed primitive root of unity of degree $p$, and let ${\mathbb K}={\mathbb Q}(\zeta)$ be the $p$th cyclotomic field. Consider the set of all non-zero ...

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606 views

### On certain representations of algebraic numbers in terms of trigonometric functions

Let's say that a real number has a simple trigonometric representation, if it can be represented as a product of zero or more rational powers of positive integers and zero or more (positive or ...

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338 views

### Some questions about the ring Z((x))

$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\dim}{\text{dim }}$
Let me begin by apologizing for the length of this question, but I thought this might be interesting to some of you. This ring isn't ...

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393 views

### For which rational values of $c$ and $d$ are the numbers $\sin{(\pi\cdot c)}$, $\sin{(\pi\cdot d)}$ and $1$ linearly dependent over $\mathbb{Q}$?

A year ago, I posted this problem on [MSE]. After a number of edits, I have arrived at the following more general problem (suggested by Hjalmar Rosengren; see the comments below).
For which ...

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**1**answer

176 views

### automorphism group of a given period

Maxim Kontsevich and Don Zagier defined the algebra of periods and conjectured that one can pass from a representation of a given period to another one using only three rules. Assuming this ...

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107 views

### Number of lattice points in a given triangle

Given a triangle with real coordinates, does anybody know how to find the number of lattice points contained within it? What if the points are only rational? I know Pick's formula can be used for the ...

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57 views

### Nontrivial norm values in rings of integers

Let $L=Q(\sqrt{d})$ for some SQF integer $d\equiv_4 3$ (the same can be asked for $d\equiv_4 1$). In this case the ring of integers of $L$ is $O_L=\{x+\sqrt{d}y \mid x,y\in Z\}$ so the norms of $O_L$ ...

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280 views

### Evaluating products of cyclotomic polynomials at roots of unity

Are there general non-trivial conditions on $p(\cdot)$ and $n$, where $p(\cdot)$ is a product of cyclotomic polynomials and $n$ is a positive integer, such that all the coefficients of $p(\cdot)$ are ...

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**1**answer

160 views

### Chebyshev polynomials factoring uniformly modulo all primes

Consider the Chebyshev polynomial of the first kind $T_n(x)$ and its factorization in $\mathbb F_p$ for a given prime $p$. Most often, this factorization is not uniform (meaning that the irreducible ...

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### Relation between cyclotomic character and fundamental character of level

The question I have is the following: is it true that the $p+1$ exponentiation of the fundamental character of level $2$ gives us the reduction (mod $p$) of the cyclotomic character?
For a review of ...

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**1**answer

92 views

### Does the Lehmer quintic parameterize certain minimal polynomials of the $p$th root of unity for infinitely many $p$?

The solvable Emma Lehmer quintic is given by,
$$F(y) = y^5 + n^2y^4 - (2n^3 + 6n^2 + 10n + 10)y^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)y^2 + (n^3 + 4n^2 + 10n + 10)y + 1 = 0$$
with discriminant $D = (7 + ...

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### examples of class fields

Can anyone explain with a numerical example of generating class field with Kummer extension? I have not come across any standard reference which does give examples. Please help or cite any reference ...

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### $\mathbb{Q}$-forms of $\mathrm{SL}_2\times \mathrm{SL}_2$

I am learning something about lattices in algebraic groups. Consider the algebraic group $\mathrm{SL}_2\times \mathrm{SL}_2$. What are the $\mathbb{Q}$-forms of such groups?

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500 views

### Adjoining torsion points from abelian varieties

Let $L/\mathbb{Q}$ be the field generated over $\mathbb{Q}$ by all of the (projective) coordinates of all of the torsion points of all abelian varieties defined over $\mathbb{Q}$. Is $L$ algebraically ...

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272 views

### How to test if the power of some algebraic number is the rational combination of two specific algebraic numbers?

Suppose we are given three algebraic numbers $\alpha,\beta,\gamma$ by presenting their minimal polynomial (degree less than $m$), the goal is to compute all positive integers $n$ such that $\alpha^n$ ...

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### Fermat's last theorem over larger fields

Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite.
Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite?
Here ...

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259 views

### On a minimal algebraic number field which satisfies the principal ideal theorem

By an algebraic number field, we mean a finite extension field of the field of rational numbers.
Let $k$ be an algebraic number field, we denote by $\mathcal{O}_k$ the ring of algebraic integers in ...

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188 views

### not Gauss sum with the same magnitude

Gauss sum is a sum of $p$ roots of unity with magnitude $\sqrt{p}$. Does another sum with such property exist?
More exactly. Let $p$ be a prime number. $\zeta^p=1,\;\zeta\ne 1$. Causs sum: ...

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70 views

### Nilpotent differential operators

I am reading Dwork's book an Introduction to G-Funcions and confronted with a problem. In section 2, chapter III (page 81), he assumes that $\mathscr F=K(X)$ the field of rational functions with ...

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447 views

### References for $K_{4k}(\mathbb{Z})$

Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be ...

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201 views

### The number of solutions of a Diophantine equation [closed]

Is $\lim_{n \rightarrow \infty} |\{(x,y) \in \mathbb{Q}(\zeta_n)^2 : y^3 = x^3 + x + 1\}| < \infty ?$ where $\zeta_n$ is a primitive $n$-th root of unity.
That is, I am asking whether the number ...

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175 views

### Diophantine equations over cyclotomic fields

Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of ...

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435 views

### Disjoint images of polynomials

Are there any $f,g \in \mathbb{Q}[x]$ such that for every root of unity $\zeta$, and every $a,b \in \mathbb{Q}(\zeta)$, $f(a) \neq g(b)?$

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### Computing local volumes : the case of Hecke p-adic subgroups

I am quite interested in knowing how to compute some volumes of groups defined on local fields $K$, mainly in order to evaluate the identity term in trace formulas. It is something well done in the ...

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878 views

### Is a number field uniquely determined by the primes which split in it?

Let $K/\mathbb{Q}$ be a number field. We say that a rational prime $p$ splits in $K$ if there exists a prime $\mathfrak{p}$ of $K$ above $p$ of interia degree $1$.
Is a number field $K$ ...

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**1**answer

62 views

### Slope decomposition of a product of operators

I'm trying to relate the slope decomposition of a product of linear operators to the slope decompositions with regard to each of the operators in the product.
First I'll give some background, for ...

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### Skew symmetry for the Hilbert symbol

Let $K$ be a local field containing the group $\mu_n$ of $n$th roots of 1 and the $\theta_K:K^*\to G_K^{ab}$ be the reciprocity map. The we know that the Hilbert symbol $$K^*\times K^*\to \mu_n$$ ...

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402 views

### Textbook request for class field theory [duplicate]

I am studying class field theory. I need good reference books, notes, or other materials which explain the following topics: ideles and ideals, Haar measure and integration on local fields, Fourier ...

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### What are the minimal degrees of the real and imaginary part of an algebraic complex number? [closed]

Let $z=a+bi\in\mathbb C$ with $b\ne0$ be an algebraic complex number of minimal degree $n$. It is obvious that $a=\dfrac {z+\bar{z}}2$ and $b=\dfrac {z-\bar{z}}{2i}$ are also algebraic. For $n=3$, it ...

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536 views

### A good book on adeles and ideles

Many results in number theory are stated either in a classical language or in an adelic one. I am often impressed of the efficiency and the satisfactory computational properties of the adelic setting, ...

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**1**answer

144 views

### What is the ring $A_{\Gamma}$ in the Cohen-Lenstra Heuristics?

I understand the work in Cohen and Lenstra's paper that leads up to the heuristics themselves, where they count weighted averages of functions defined over isomorphism classes of $A$-modules, where ...

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195 views

### Intersection of a ring class field of a quadratic field K with the cyclotomic extension of K

Let $K$ be a quadratic field. Let $f\in\mathbb{Z}_{\geq 1}$. Let $\mathcal{O}_f=\mathbf{Z}+f\mathcal{O}_K$ be the unique order of $K$ of index $f$ in $\mathcal{O}_K$. Let $H_f^{ring}$ denote the ring ...

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### What is an infinite prime in algebraic topology?

The links between algebraic topology (stable homotopy theory in particular) and number theory are nowadays abundant and fruitful. In one direction, there is chromatic homotopy theory, exploiting the ...

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### Rings of algebraic integers as quotients of polynomial rings

The ring of integers $\mathcal{O}_K$ of a number field $K$ is always isomorphic to some ring of the form $\mathbb{Z}[x_1, ..., x_r]/\mathfrak{p}$, where $\mathfrak{p} \subset \mathbb{Z}[x_1, ..., ...

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### Reciprocity laws in different dimensions

Let $M/L/Qp$ be a finite galois abelian extension of local fields and define
$\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathbb{Z}}a_iT^i:a_i\in M,\min_{i\in \mathbb{Z}}, v(a_i)>−\infty , \lim_{i\to ...

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### Relation between 1-dimensional and 2-dimensional reciprocity maps

Let $M/L/\mathbb{Q}_p$ be a finite galois abelian extension of local fields and define
$\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathcal{Z}}a_iT^i : a_i\in M, \min_{i\in \mathcal{Z}} v(a_i)>-\infty, ...

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### The construction of the 257gon

If $\zeta\in\mathbb C$ is a primitive $257$th root of unity, the Galois group $\operatorname{Gal}(\mathbb Q(\zeta)/\mathbb Q)$ is cyclic of order $256=2^8$, so we know that there is a sequence of $8$ ...

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### Iwasawa's mu-invariant for noncyclotomic $\mathbf{Z}_p$ extensions of cyclotomic fields?

Let $p$ be an odd prime number, $m$ a positive integer with $p\mid m$. Put $k=\mathbf{Q}(\mu_m)$.
(1) Is there any example where certain noncyclotomic $\mathbf{Z}_p$-extension $k_\infty/k$ has ...

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### Lowest degree polynomial with integer coefficients yielding $1/\sqrt{2^n}$

Let $x = \cos(\pi/8) = \frac{1}{2} \sqrt{2+\sqrt{2}}$ and $y = \sin(\pi/8) = \frac{1}{2} \sqrt{2-\sqrt{2}}$. What is the lowest degree polynomial $p(x,y)$ with integer coefficients such that $p(x,y) = ...

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### Units of an extension of $\mathbb{Z}$ [closed]

Let $P(x)\in\mathbb{Z}[x]$ be monic and irreducible over $\mathbb{Q}[x]$, and let $\theta$ be a root of $P(x)$. Let $K = \{a + b\theta\} \subseteq \mathbb{Z}[\theta]$. When is it the case that there ...

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### Galois groups and prescribed ramification

What is known about finite groups $G$ for which there exists a Galois extension $K$ of $\mathbb{Q}$ ramified only at $2$ such that $\text{Gal}(K/\mathbb{Q}) \cong G$ ? More generally, which groups can ...

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380 views

### How did height in algeb. number theory/elliptic curves started?

Maybe this is obvious but it isn't to me yet. What is the history of heights used in say points of the project plane over a number field or of elliptic curve over a number field? I would guess people ...

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154 views

### Motivating mathematics(particularly algebraic number theory) through historical problems [closed]

Most mathematical textbooks start a subject by going backwards, historically. They will define the terms that were invented to solve a problem in their polished form and then use these definitions and ...

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113 views

### Hilbert Class fields and Pure cubic fields

I want to know about Hilbert class field theory for pure cubic fields. Which is the best source for this? Right now I am reading book by D.A. Cox "Primes of the form $x^2+ny^2$" Fermat,Class field ...