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4
votes
1answer
134 views

A strange condition on containment of special complex numbers in cyclotomic fields

In a recent theorem we have naturally come across this condition, that seems to be important, but rarely satisfied: $\sqrt{\frac 1 4 + a^m} \in \mathbb Q(\zeta_m, a)$ where $a\in\mathbb C^*$ and ...
-2
votes
0answers
53 views

the special value in a L-funtion [on hold]

how to calculate the value of series The Gamma function satisfying $\Gamma(n) = (n-1)!\quad\forall n\in\mathbb N$ is via through the Euler integral $$ \Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt\,. $$ ...
0
votes
0answers
17 views

How to character the norm of elemental units in a quadratic number field [on hold]

if a prime number P≡1 mod 4,why the equation x^2-py^2=-4 has solutions in integer
3
votes
1answer
278 views

UFD and fundamental group

Let $C$ be the curve $x^2+y^2-1$, defined over $\mathbb R$. It is easy to see that $\mathbb R[C]$ is not a UFD, as witnessed by the identity $(1-x)(1+x)=y^2$. On the other hand, the real locus ...
0
votes
0answers
99 views

Solving the transcendental equation $Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$

I need to solve the following equation: $Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$ for $x\in\mathbb{R}^{\ast}$ and where $k\in\mathbb{R}^{+}$. Here $Li_{3}$ and $Li_{2}$ are the third and ...
0
votes
0answers
44 views

Determinant of a Vandermonde matrix of roots of monic polynomial with integer coefficients [migrated]

Let $p(x)=\sum_{i=1}^n a_ix^i$ with $a_i$ an integer for all $i$ and $a_n=1$ such that $p(x)$ has only real roots, and let $\lambda_1,\ldots,\lambda_n$ be the $n$ roots of this polynomial. Then the ...
3
votes
0answers
164 views

Density of primes of degree one in Bauer's Theorem (Application of Chebotarev Density)

Let $L$ be a Galois extension of $\mathbb{Q}$ and $M$ a finite extension of $\mathbb{Q}$, both of degrees $> 1$. A Theorem of Bauer tells that $Spl_1(M)\subset Spl(L)$ up to a finite number of ...
6
votes
2answers
392 views

How to compute class number of a torus

Let $T$ be an algebraic torus over a number field $K$. Following notations in Ono's The Arithmetic of Tori, ...
2
votes
0answers
99 views

algorithm to find a new point of small height in a number field extension

By the height of an algebraic number $\alpha$, I mean the absolute, logarithmic (additive) Weil height $h(\alpha)$; e.g. $h(2^{1/n}) = (\log 2)/n.$ If $K$ is a number field, let $\delta(K)$ denote ...
1
vote
0answers
113 views

Unramified extensions of a given degree

Let $K \neq \mathbb{Q} $ be a finite extension of $\mathbb{Q}$. For a given integer $n$, how to construct an unramified extension of $K$ of degree $n$ ? EDIT: If not then under what conditions on ...
0
votes
1answer
83 views

A particular argument in the review on expanders by Hoory-Linial-Wigderson

I am thinking about the third bullet point on page 455 here, http://www.ams.org/journals/bull/2006-43-04/S0273-0979-06-01126-8/ Can someone explain what is the argument there which seems to conclude ...
5
votes
1answer
161 views

Constructing Ramanujan graphs from elliptic curves

Is there an exposition which explains how to do this step-by-step? (I see stray references and allusions to such a thing being possible but can't locate anything concretely) Something to do with ...
3
votes
1answer
196 views

Dihedral extension of 2-adic number field

Sorry if the question is too long and maybe elementary. I am reading a paper by Hirotada Naito on "Dihedral extensions of degree 8 over the rational p-adic fields". To generate dihedral extension ...
5
votes
0answers
118 views

Degenerate linear recurrence sequences

Let $(u_n)_{n \geq 0}$ be a linear recurrence given by $$u_n = a_1 u_{n-1} + \cdots + a_k u_{n-k} \quad \forall n \geq k ,$$ where $u_0, \ldots, u_{k-1}, a_1, \ldots, a_k \in \mathbb{Z}$. We recall ...
5
votes
1answer
425 views

Group laws in class field theory

In the case of a quadratic imaginary number field one can construct its maximal abelian extension using torsion points of an elliptic curve with complex multiplication by this field. In the case of a ...
0
votes
0answers
58 views

separating parameters in generalized quadratic Gauss sum

The normalized generalized quadratic Gauss sum is defined by $$ G(a,b,c)=\frac{1}{c}\sum_{n=1}^ce\left(\frac{an^2+bn}{c}\right) $$ where $e(x)=\exp(2\pi ix)$. Under what conditions on $c$ can we ...
3
votes
0answers
121 views

Application of Stickelberger's Theorem to Quadratic field

I am trying to understand a proof of the Kronecker-Weber Theorem by Franz Lemmermeyer,[http://arxiv.org/pdf/1108.5671.pdf] in which he uses Stickelberger's Theorem applied to Kummer extensions. I can ...
1
vote
0answers
110 views

Averages of $L(s,\chi)$

Let $(\frac{m}{n})$ denote the usual quadratic Jacobi symbol. What is the abscissa of convergence of the double Dirichlet series ? $$ \sum_{\substack{m,n \in \mathbb{N} \\ \gcd(m,n)=1 \\m,n\equiv 1 ...
2
votes
2answers
175 views

Some general properties of arithmetic groups of simplest type

I'm working in the area of arithmetic Kleinian groups (as discrete groups of motions of hyperbolic 3-space). For the more general case of hyperbolic $n$-space, there is a particular class of ...
2
votes
1answer
133 views

Irreducibility of Faulhaber-like Polynomials over $\mathbb Q[x]$

Motivation: Inspired by the famous Faulhaber polynomials $F_k(N)=\displaystyle\sum_{n=0}^Nn^k,$ I decided to study their alternating versions, $\Phi_k(M)=\displaystyle\sum_{n=0}^M(-1)^nn^k$. For ...
4
votes
1answer
277 views

Square-free grows as $6n/\pi^2$: $k$-th free?

The asymptotic number of square-free numbers $\le n$ is $Q(n) = 6n/\pi^2 + O(\sqrt{n})$. Because $\zeta(2)=\pi^2/6$, $Q(n) \approx n/\zeta(2)$. OEIS A004709 says that cube-free numbers have ...
0
votes
1answer
103 views

Irreducibility of cyclotomic polynomial over real quadratic number field

Let $n$ and $d$ be positive integers, with $d\ge 2$ square-free. It is well known that $\Phi_n=\Phi_n(x)$, the $n$-th cyclotomic polynomial, is irreducible over $\mathbb{Q}$. However, as the simple ...
8
votes
2answers
701 views

What is known about primes of the form x^2-2y^2?

David Cox's book Primes of The Form: $x^2+ny^2$ does a great job proving and motivating a lot of results for $n>0$. I was unable to find anything for negative numbers, let alone the case I am ...
1
vote
0answers
73 views

Spectrum of primitive nonnegative integer matrices

Let $P(X) = a_nX^n + \cdots + a_1X + a_0$ with $a_i \in \mathbb Z$. Question 1. Is there an efficient criterion on the $a_i$ to decide if there exists a primitive nonnegative integer matrix with ...
6
votes
1answer
141 views

Finite Nontrivial Unramified Towers of Number Fields

Let $F$ be a number field and $L=F^{un}$ its maximal unramified extension. By Class Field Theory, $$Gal(L/F)^{ab}\cong Cl(F).$$ It's well-known that we can have $[L:F]=1$ (e.g. $F=\mathbb{Q}$), and ...
8
votes
0answers
316 views

What is known about the reverse mathematics of algebraic number fields?

I know work on the reverse mathematics of countable algebraic field extensions including Galois theory, notably including Dorais, Hirst, and Shafer http://arxiv.org/pdf/1209.4944v2.pdf. But algebraic ...
14
votes
0answers
377 views

Can there be arbitrarily many cubic fields unramified outside $\{p,\infty\}$?

Observe, trivially, that since quadratic fields correspond to rational integers modulo squares (viz. discriminants), there are (roughly about, but certainly at most) $2^{|S|+1}$ quadratic fields ...
12
votes
2answers
583 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 ...
18
votes
1answer
649 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 ...
0
votes
1answer
116 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 ...
6
votes
1answer
405 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 $\epsilon_1 + \epsilon_2 = 1$ ...
8
votes
2answers
339 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 ...
2
votes
0answers
90 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 ...
22
votes
0answers
643 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 ...
6
votes
1answer
366 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 ...
5
votes
0answers
404 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 ...
1
vote
1answer
179 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 ...
0
votes
0answers
122 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 ...
0
votes
0answers
69 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 $O_L$ ...
11
votes
0answers
285 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 ...
2
votes
1answer
167 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 ...
0
votes
0answers
96 views

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 ...
2
votes
1answer
96 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 + ...
0
votes
0answers
113 views

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 ...
2
votes
0answers
107 views

$\mathbb{Q}$-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?
8
votes
2answers
510 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 ...
4
votes
1answer
387 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 $\alpha^n$ ...
33
votes
4answers
3k views

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 ...
7
votes
2answers
274 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 ...
2
votes
1answer
189 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: ...