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5
votes
0answers
62 views

Ideal classes fixed by the Galois group

Let $K$ be a number field and let $G$ be the group of automorphisms of $K$ over $\mathbf Q$. The group $G$ acts in a natural way on the ideal class group of $K$. I would like to know if there are any ...
0
votes
0answers
50 views

Multiplicativity of the Ideal Norm

I asked this question on StackExchange twice, and was never able to get an answer. This is not quite a research-level question, but it is sufficiently difficult and interesting from a pedagogical ...
11
votes
2answers
354 views

Asymptotics for algebraic numbers of height less than one

The question. Is an asymptotic equivalent known or conjectured for the number $N(d)$ of $\alpha \in \bar{\mathbb{Q}}$ with $h(\alpha) < 1$ and $[\mathbb{Q}(\alpha):\mathbb{Q}] \leq d$? The rather ...
0
votes
0answers
127 views

A question on linear forms

Let $K/\mathbb{Q}$ be a fixed number field of degree $n$, and let $\sigma_1, \cdots, \sigma_n$ denote the distinct embeddings of $K$ into $\mathbb{C}$. Suppose that $\alpha_1, \cdots, \alpha_{r}$ are ...
5
votes
0answers
182 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
457 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 ...
0
votes
0answers
415 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?
5
votes
2answers
274 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 ...
4
votes
1answer
189 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
86 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 ...
1
vote
2answers
135 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 ...
5
votes
1answer
276 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 ...
3
votes
0answers
231 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 ...
6
votes
0answers
383 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 ...
1
vote
0answers
43 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
505 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
151 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
345 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 ...
3
votes
0answers
58 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
386 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, ...
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
154 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 $p$.) ...
24
votes
0answers
327 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 ...
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 ...
13
votes
2answers
466 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
308 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, ...
31
votes
1answer
868 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
208 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.
8
votes
2answers
455 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 ...
5
votes
1answer
273 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 ...
4
votes
1answer
219 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*} ...
2
votes
2answers
392 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 ...
14
votes
1answer
732 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 ...
4
votes
2answers
127 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 ...
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 ...
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
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 ...
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 ...
11
votes
3answers
242 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 ...
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 ...
7
votes
1answer
360 views

Hilbert Class Field Galois over Q?

So if we have a Galois extension $K/\mathbb{Q}$, then the Hilbert Class Field $H$ of $K$ is certainly Galois over $\mathbb{Q}$. But is the converse true? I know many examples of nongalois ...
-1
votes
1answer
141 views

Irreducible polynomial on $\mathbb{F}_{2}[x]$

For some reason I need some irreducible polynomial $f$ on $\mathbb{F}_{2}[x]$ where $\deg f \in [10^3,10^6]$. Could someone give information about this? Thx.
3
votes
3answers
329 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 ...
2
votes
0answers
115 views

Domains with prime ideal theorems

Let $D$ be a domain, and for prime ideals $\frak P$ of $D$ the norm is $N({\frak P}):=|D/{\frak P}|$. The prime ideal counting function of $D$ is given by $\pi_D(x)=\#\{{\frak P}\in{\rm ...
9
votes
1answer
662 views

How small can a totally positive integer be?

Consider a large, fixed $M>2$. For each $n$, let $\alpha_n$ denote the smallest algebraic integer of degree at most $n$, all of whose Galois conjugates lie in the real interval $(0,M)$. Is there ...
1
vote
1answer
185 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 ...
3
votes
1answer
170 views

A question on the genus field of an algebraic number field

The following is quoted from the Mathematical Reviews. MR0544896 (80j:12002) Reviewed Bhaskaran, M. Construction of genus field and some applications. J. Number Theory 11 (1979), no. 4, 488–497. ...