The tag has no wiki summary.

learn more… | top users | synonyms

4
votes
0answers
99 views

Factors of the polynomial $X^n-a$

I am interested in the polynomial $X^n-a$ in $\mathbb{Q}[X]$, for some $a\in \mathbb{Q}^*$, and would like to know the irreducible factors of it. Is there something in the literature which gives a ...
6
votes
3answers
417 views

Constructing quintic number fields with certain splitting behaviour

I am looking for number fields $K$ which satisfy the following properties: $[K:\mathbb{Q}]=5$. The Galois closure of $K$ has Galois group $S_5$. For each prime $p$ which ramifies in $K$, there ...
4
votes
2answers
257 views

Time-line until the publicaton of Weil of “Numbers of solutions of equations in finite fields”

In "On the history of the Weil Conjectures" Dieudonné says: "Appropriately enough, the story, as with so many problems in number theory, begins with Gauss...". C. F. Gauss, Disquisitiones ...
0
votes
0answers
403 views

A letter from J. P. Serre

Which is the letter where J. P. Serre present "Analogues Kählériens de certaines conjectures de Weil" to Weil?
-1
votes
0answers
75 views

Help with extension of f (mentioned below) to f: Zp -> Zp ,continuous function [on hold]

This is a (different version to) question from Serre 'A Course in Arithmetic'.Let p be an odd prime number. $\forall n\geq 1$ (n positive integer), $f$ is defined by: $$f(n)=(-1)^n\prod_{1\le k\le n ...
4
votes
1answer
177 views

How frequently is 3 a cubic residue mod primes in an arithmetic progression?

Suppose $(a,3q)=1$ and $a\equiv 1\pmod 3$. Are there infinitely many primes $p\equiv a\pmod {3q}$ such that $3$ is a cubic nonresidue modulo $p$? Or, an equivalent formulation using quadratic forms: ...
2
votes
1answer
82 views

Why do noncocompact arithmetic Kleinian groups have quadratic trace fields?

I realize there are a few different ways of going about proving this, depending on one's background, but there's a particular number theoretic aspect that I am just blanking on, and can't seem to find ...
11
votes
3answers
240 views

Origin of number theoretic invariants associated to hyperbolic 3-manifolds

I've been studying number theoretic methods of classifying hyperbolic 3-manifolds for over a year now. In particular, there is are the trace field, invariant trace field, quaternion algebra, and ...
1
vote
2answers
130 views

On the conductor of the Groessencharacter of a CM elliptic curve

Let $K$ be a quadratic imaginary field. Let $L$ be a number field which contains $K$ and let $E/L$ be an elliptic curve defined over $L$ with complex multiplication by $K$, i.e. such that ...
3
votes
0answers
223 views

Independent units in pure number fields $\mathbb{Q}(\sqrt[p]{t})$

Theorem: In this paper of Frei and Levesque, they correct the proof of a result of Halter-Koch and Stender: Define the real pure algebraic number field $\mathbb{K}=\mathbb{Q}(\omega_n)$ for ...
1
vote
1answer
627 views

Countable sets of permutations of an infinite binary sequence

Being motivated by this recent post of mine, I am thinking about the base 2 decimal expansion of irrational algebraic numbers. My main question is whether or not the reasoning I outline below ...
1
vote
1answer
231 views

Number Field Sieve for factorization with non-monic non-linear polynomial. Can't understand calculating ideal valuations

There is a paper "Factoring integers with the number field sieve" (download it here, for example). I can't understand how they reason the correctness of computing ideal valuations in the case of ...
5
votes
0answers
280 views

Is formula valid for relating $\pi$ with ALL of its OEIS A002485(n)/A002486(n) convergents?

Could anyone try to prove that the below conjectured formula is valid for relating $\pi$ with ALL of its convergents - those, which are described in OEIS via A002485(n)/A002486(n) ? $$(-1)^n\cdot(\pi ...
5
votes
1answer
266 views

Connection of Galois representation and arithmetic geometry

This is might be a dumb question. There are lots of Galois representations which arise naturally from geometric objects, for example, Galois representations attached to elliptic curves. I know that ...
8
votes
2answers
348 views

Class number of real maximal subfield of cyclotomic fields

Let $p$ be a prime number and $h_p^+$ the class number of $\mathbb{Q}(\zeta_p + \zeta_p^{-1})$. What is known about the values of $p$ for which $h_p^+ = 1$? Are there infinitely many? Finitely many? ...
1
vote
0answers
41 views

What is the explicit eigenvalues of Hilbert modular forms?

Let $F$ be a totally real number field and let $I$ denote the set of embeddings $\tau:F\to \mathbb{R}.$ Let $k=(k_\tau)\in\mathbb{Z}^I_{>0}$ and suppose all the $k_\tau$'s have the same parity. Let ...
5
votes
2answers
499 views

Conjecture on irrational algebraic numbers

Conjecture: For every irrational algebraic number $q$ and natural number $b$, the representation of $q$ on base $b$ contains all the digits $[0,\dots,b-1]$. Questions: Has this conjecture been ...
2
votes
0answers
147 views

closed form expression for an infinite series

Is there any closed form expression for the infinite sum $\sum_{n \geq 0}q^{n(n+1)/2}(1+q)(1+q^2)\cdots(1+q^n)u^n$ where both $q$ and $n$ are variables and $n \in N \cup {0}$?
1
vote
2answers
339 views

Chinese Remainder Theorem backwards

I have the following situation, that is much alike the Chinese Remainder Theorem. Let $\phi_d(\alpha)$ be the $d^{th}$ cyclotomic polynomial in the variable $\alpha$ (I'm not specifying the ...
6
votes
2answers
380 views

Elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$

For some research work, I need to know the classification of elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$, up to conjugation. Since I essentially need $n\le 4$, I think that I can show it ...
3
votes
0answers
57 views

Iwasawa theory: Do these $\mu$-invariants of a number field coincide?

Let $k$ be a number field and let $k_\infty$ be the cyclotomic $\mathbf{Z}_p$-extension of $k$. Put $\Gamma=G(k_\infty/k)\cong \mathbf{Z}_p$, $\Lambda=\mathbf{Z}_p[[\Gamma]]$. Let $S$ be a finite set ...
4
votes
3answers
377 views

Class number for binary quadratic forms discriminant $\Delta$ to class number $\mathbb Q(\sqrt \Delta)$

Jyrki Lahtonen has suggested I write a blog post relating binary quadratic forms to quadratic field class numbers, ...
24
votes
0answers
696 views

Derivative of Class number of real quadratic fields

Let $\Delta$ be a fundamental quadratic discriminant, set $N = |\Delta|$, and define the Fekete polynomials $$ F_N(X) = \sum_{a=1}^N \Big(\frac{\Delta}a\Big) X^a. $$ Define $$ f_N(X) = ...
2
votes
1answer
310 views

Finding a suitable number

Let $n,m$ be two positive integers. By $r_n$ we denote the largest prime not exceeding $n$. If $r_n\leq m\leq n$ and $q$ is the largest prime factor of $n!/m!$ such that $q\geq 17$ and $q\geq n-m+3$, ...
1
vote
0answers
152 views

Unique factorization in a quadratic extension of $Z_p$ [closed]

Let $\mathbf{Z}_{(p)}$ denote the ring of all rational numbers whose denominators in lowest terms are not divisible by the integer prime $p$. (This is normally described as the localization at $p$.) ...
2
votes
2answers
388 views

GL(2) Local Langlands and Artin's L-function

The context I am thinking of mainly is GL(2), and accordingly, the degree 2 Artin L-function. But comments about GL(n) in general are also welcome. In light the local Langlands correspondence, what ...
24
votes
0answers
319 views

Degree 17 number fields ramified only at 2

The number $17$ is the smallest odd number that occurs as the degree of a number field $K/\mathbb{Q}$ for which the only finite prime that ramifies is $2$. The non-existence for $n < 17$ follows ...
13
votes
2answers
451 views

Are there any simple, interesting consequences to motivate the local Langlands correspondence?

Let's pretend that we know local Langlands at a fairly high level of generality... i.e. we know something along the lines of: Let $G=\mathbf{G}(F)$ be the group of $F$-points of a connected ...
7
votes
1answer
307 views

Positive primes represented by indefinite binary quadratic form

Neil Sloane asked me about commands in computer languages to find the (positive) primes represented by indefinite binary quadratic forms. So I wrote something in C++ that works. This is for the OEIS, ...
5
votes
1answer
152 views

Positive Primes represented by an indefinite binary form, reducing poly degree from 8 to 4

In his lovely answer at Positive primes represented by indefinite binary quadratic form Noam found that a (positive) odd prime $p$ is represented by the indefinite form $x^2 + 13 x y - 9 y^2$ if and ...
4
votes
2answers
126 views

unit group of biquadratic fields

In the unit group of a real biquadratic field, what is the index of the product of the unit groups of its three quadratic subfields? Is the index 1 if discriminant of these three subfields are always ...
1
vote
0answers
207 views

Where can I find the article of A. Borel: “Values of zeta-functions at integers, cohomology and polylogarithms”? [closed]

Where on the internet can I find this article? I know that it is in this book: Current trends in mathematics and physics, Narosa, New Delhi, 1995.
31
votes
1answer
866 views

Is my field algebraically closed?

For a field $L$, let $\widetilde L$ be the splitting field of all irreducible polynomials over $L$ having prime-power degree. Question: Do we have $\widetilde{\mathbf Q}=\overline{\mathbf Q}$? ...
1
vote
0answers
102 views

Isogenies in multidimensional formal groups

Let $K/\mathbb{Q}_p$ be a local field, $A$ the ring of integers of K, $\pi$ a uniformizer element for $A$, $F$ an n-dimensional formal group with coefficients in $A$ and $f$ an endomorphism of $F$. ...
4
votes
1answer
216 views

Examples of specializations of elementary symmetric polynomials

Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$ indeterminates. The $h^{th}$elementary symmetric polynomial is the sum of all monomials with $h$ factors \begin{eqnarray*} ...
5
votes
1answer
271 views

Neukirch's papers and theorem

Have any of Neukirch's papers on anabelian geometry been translated? I'm mostly interested in: Kennzeichnung der p-adischen und der endlichen algebraischen Zahlkörper (1969) Kennzeichnung der ...
18
votes
1answer
312 views

Proportion of irreducible polynomials $P$ such that $\mathbf Z[X]/(P)$ is the ring of integers of $\mathbf Q[X]/(P)$

I know that number fields have been the object of many statistical experiments. Is there some kind of heuristics for the following? Fix a degree $d$ and fix a bound $N$ on the coefficients of a monic ...
6
votes
1answer
237 views

Degree of Kummer extensions of number fields

Let $K$ be a number field and $a\in K^*$ of infinite order in $K^*$. How do I show that $$[K(\sqrt[n]{a},\zeta_n):K]\geq C\cdot n\cdot\varphi(n)$$ holds for all positive integers $n$, with a positive ...
14
votes
1answer
729 views

Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,…)

Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ... On the other hand model theory, in particular after Hrushovski, found many ...
3
votes
3answers
328 views

Distinct primitive factorizations over integers of number fields

I am curious about the following. Let $K$ be a number field. For any $a \in \mathcal{O}_K$ in its ring of integers, let $N(a)$ be zero if there exist elements $b, c \in \mathcal{O}_K \setminus ...
20
votes
0answers
912 views

Orders in number fields

Let $K$ be a degree $n$ extension of ${\mathbb Q}$ with ring of integers $R$. An order in $K$ is a subring with identity of $R$ which is a ${\mathbb Z}$-module of rank $n$. Question: Let $p$ be an ...
1
vote
1answer
111 views

Quotients of number rings IZ[zeta_l]

Let $l=p^r$ a prime power and $\zeta$ a primitive l-th root of unity. It is classical result, that $(1-\zeta)^{\varphi(l)}=p\cdot\epsilon\in\mathbb{Z}[\zeta]$ for a unit $\epsilon$. It should be a ...
4
votes
2answers
191 views

Orders of the conjugates of an algebraic prime number in its residue field

Of interest to me is the following question (it would be nice to find out what is known in its direction): Given a Galois number field $K/\mathbb{Q}$ and a completely and principally split prime ...
4
votes
2answers
272 views

When does a dyadic prime ramify in a relative quadratic extension?

In a quadratic extension $\mathbb{Q}(\sqrt{d})$of $\mathbb{Q}$ it is clear that 2 ramifies if and only if $d\equiv 2,3\mod 4$ (easy to see if you compute the discriminant). But if I take a relative ...
8
votes
1answer
246 views

Reference for the fact that $SL_n(O_K)$ surjects onto $SL_n(O_K/I)$ for any ideal I

Let $\mathcal{O}_K$ be the ring of integers in an algebraic number field $K$ and let $I \subset \mathcal{O}_K$ be a nonzero proper ideal. It is not hard to see that the map ...
1
vote
1answer
182 views

Arithmetic property of a surface of general type

In my previous post I asked about the hyperbolicity of the affine surface $S'=\{zw \neq o\}$ in the projective surface $z^2 = P(x) Q(y)$ in $\mathbb{P}^3$, where $P$ and $Q$ are two general ...
0
votes
0answers
57 views

Programs to find class number of bi-quadratic fields

I'm interested in studying ideal class group structure for bi-quadratic fields. The problem I'm facing is the lack of examples. Is there any program where I can find the class number of a bi-quadratic ...
0
votes
0answers
50 views

Extensions on Higher-dimensional local fields

I have the following question: Let $M/L$ b a finite extension of n-dimensional local fields and $t_1,\dots, t_n$ a system of local parameters of $L$ with valuation $v$. Let us fix an $1\leq i \leq ...
1
vote
0answers
36 views

Valuations in Higher-dimensional local fields

I have the following question which I believ should be true but I would like to have a different opinion about it: Let $M/L$ is a finite Galois extension of $n$-dimensional local fields and ...
0
votes
0answers
68 views

primitive polynomial on $\mathbb{F}_2[x]$

For some reason I need some primitive polynomial $f$ on $\mathbb{F}_2[x]$ where $\deg f \in [1,10^4]$. (Especially for $\deg f = 10\pm \epsilon, 10^2\pm \epsilon, 10^3\pm \epsilon, 10^4 \pm ...