# Tagged Questions

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 ...

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### divisible by all standard prime numbers

This question is about prime numbers in nonstandard models of Peano Arithmetic. Every such model looks like N+AxZ, where A is a dense linear order without end points. There are many nonstandard ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 \... 2answers 242 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 ... 1answer 220 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 k\... 1answer 325 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 ... 1answer 140 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 ... 2answers 1k 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 ... 0answers 101 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 ... 0answers 238 views ### Analog of Euler's factoring technique Is there an analog of Euler's Two Squares factoring theorem over polynomial rings \Bbb Z[x] by considering a version for non-negative polynomials? Euler's two squares factoring states that numbers ... 1answer 188 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 [... 0answers 497 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 ... 0answers 469 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 ... 2answers 797 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 ... 1answer 733 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 pth root of unity will do as well). For ... 1answer 150 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 ... 1answer 439 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 ... 2answers 474 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 ... 0answers 102 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 pth cyclotomic field. Consider the set of all non-zero ... 0answers 706 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 ... 1answer 608 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 ... 0answers 430 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 ... 1answer 201 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 conjecture,... 0answers 344 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 ... 0answers 162 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 ... 0answers 325 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 ... 1answer 199 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 ... 0answers 157 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 ... 1answer 124 views ### Does the Lehmer quintic parameterize certain minimal polynomials of the pth 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 $D = (7 + ... 0answers 124 views ### examples of class fields Can anyone explain with a numerical example of generating class field with Kummer extension? I have not come across any standard reference which does give examples. Please help or cite any reference ... 0answers 118 views ###$\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}$-forms of such groups? 2answers 550 views ### Adjoining torsion points from abelian varieties Let$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$algebraically ... 1answer 495 views ### How to test if the power of some algebraic number is the rational combination of two specific algebraic numbers? Suppose we are given three algebraic numbers$\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$... 4answers 3k views ### Fermat's last theorem over larger fields Fermat's last theorem implies that the number of solutions of$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}}$finite? Here$\...
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 $k$...