The algebraic-number-theory tag has no wiki summary.

**1**

vote

**0**answers

168 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 $p$.) ...

**35**

votes

**1**answer

1k 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

**1**answer

191 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 ...

**16**

votes

**2**answers

695 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 ...

**8**

votes

**2**answers

548 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

**1**answer

991 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

**0**answers

225 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.

**9**

votes

**4**answers

819 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

**1**answer

358 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 ...

**21**

votes

**1**answer

422 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

**1**answer

257 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

**1**answer

464 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

**3**answers

2k 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

**2**answers

217 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

**2**answers

213 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

**1**answer

266 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

**1**answer

115 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

**0**answers

64 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

**3**answers

310 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

**0**answers

42 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

**0**answers

81 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 ...

**8**

votes

**2**answers

586 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

**1**answer

146 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.

**4**

votes

**3**answers

400 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

**0**answers

151 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

**1**answer

692 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

**1**answer

200 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

**1**answer

203 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.
...

**1**

vote

**1**answer

177 views

### Etale cohomology and restricted direct product

[migrated from math.stackexchange: http://math.stackexchange.com/questions/727896/etale-cohomology-and-restricted-direct-product]
$\newcommand{\h}{\operatorname{H}}$
Let $k$ be a global field, $A$ an ...

**3**

votes

**3**answers

507 views

### Non existence of cyclic infinite linear algebraic groups

Let $G$ be a linear algebraic group defined over some algebraically closed field $\mathbb{K}$ and also over some subfield $k\subset \mathbb{K}$. There is thus a natural group structure on the set of ...

**3**

votes

**2**answers

173 views

### Linear polynomials in units of number fields

I would be thankful for a reference to any result that says "how often" an equation of the form $$c_1x_1 + c_2x_2 + ... + c_nx_n = 0,$$
where $n$ is fixed, $c_1, ..., c_n \in \mathcal{O}_K$ are ...

**1**

vote

**1**answer

219 views

### Small generators of the ideal class group

If $K$ is a number field, a result from Bach tells us that the primes in $K$ of norm smaller that $12 (\log |\mathrm{Discriminant}(K)|)^2$ generate the ideal class group $\mathrm{Cl}_K$. Is there any ...

**1**

vote

**0**answers

189 views

### Discriminant of a compositum of number fields, a bound?

Given two number fields $E$ and $F$, is there a bound on $|d_{EF}|$, the absolute value of the absolute discriminant of the compositum of fields $EF$, in terms of $d_E$, $d_F$, and the extension ...

**4**

votes

**0**answers

147 views

### Continuity of the Hilbert pairing

I would like to know if the Kummer pairing (or the analogue of the Hilbert Symbol) for a one dimensional group defined over the ring of integers of a higher-dimensional local field is continuous (with ...

**7**

votes

**1**answer

313 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 ...

**5**

votes

**2**answers

346 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 ...

**2**

votes

**1**answer

86 views

### Principally split primes with factors in arbitrarily small angular sectors

I wonder if the following is known:
let $n$ be a (square-free) positive integer. Is there ever/always a sequence of prime numbers $p$ that can be written in the form $$p = x^2 + ny^2,$$ where
$x, y$ ...

**1**

vote

**0**answers

169 views

### $\mathfrak{q}$-ideal class bound

Let $K$ be a number field, $\mathcal{O}_K$ be its ring of integers. Let $\mathfrak{q}$ be a nonzero ideal in $\mathcal{O}_K$.
The $\mathfrak{q}$-ideal class group consists of equivalence classes of ...

**8**

votes

**1**answer

369 views

### Explicit isomorphism for quaternion algebras over $\mathbb{Q}$?

It is known that the isomorphism class of a quaternion algebra $A=\binom{a,b}{K}$ over a number field $K$ is determined by the finite set of places $v$ of $K$ where $A\otimes_K K_v$ is a division ...

**1**

vote

**0**answers

80 views

### Units in residue classes

Let $K$ be a CM-field of degree $2n$. (Quadratic extension of totally real number field)
Let $\mathcal{O}_K$ be the ring of integers in $K$, and $m\geq 1$ an integer. Let $U_K$ be the group of units ...

**1**

vote

**1**answer

70 views

### Integral elements of quaternion algebras with predescribed properties

In the course of doing some calculations I have found myself wanting to answer the following question:
Let $D/\mathbb{Q}$ be a quaternion algebra ramified at a prime $p$ and at $\infty$ and let ...

**1**

vote

**2**answers

195 views

### Double Density Theorem?

A colleague asks me the following: "I wonder if you can give me a reference - or a guidance where to look – from a fact I recall from graduate school. I’m sure it can be generalized quite a bit but ...

**2**

votes

**2**answers

399 views

### Connection between quadratic forms and ideal class group

I'm studying the classic results on binary (integer) quadratic forms and I'm looking for a reference on the following result (maybe a book that contains a proof):
Let $O_k$ be the ring of algebraic ...

**11**

votes

**4**answers

621 views

### Analogy between the nodal cubic curve $y^2=x^3+x^2$ and the ring $\mathbb{Z}[\sqrt{-3}]$?

I'm trying to motivate a bit of algebraic geometry in an abstract algebra course (while simultaneously trying to learn a bit of algebraic geometry), and I thought that it might be nice to present an ...

**0**

votes

**0**answers

108 views

### Cubic field and the corresponding cubic binary form

I am currently reading about binary cubic forms and cubic number fields (mainly about using binary cubic forms with integer coefficients to parametrize orders in the cubic field) and I thought it ...

**14**

votes

**1**answer

590 views

### Chebotarev density theorem for $k$-almost primes

Consider a finite Galois extension $L$ of $\mathbb Q$, of Galois group $G$. Let $k \geq 1$ be a fixed integer. Let $D$ be a subset of $G^k$ invariant by conjugation and by the natural action of the ...

**0**

votes

**1**answer

78 views

### Significance of the sign of the field norm for units in real quadratic fields

Let $k = \mathbb{Q}(\sqrt{m})$, where $m \equiv 1 \pmod{8}$. Let $\epsilon$ be the fundamental unit of $k$ satisfying $\epsilon > 1$.
A paper I'm reading involves studying the 2-torsion fields ...

**7**

votes

**2**answers

1k views

### Linear independence of the square roots over Q

Does there exist a real number $a$ such that the numbers $\sqrt{n^2 + a^2}$ (for all natural $n$) are linearly independent over the field of rational numbers? It is evident that $a$ cannot be ...

**1**

vote

**1**answer

171 views

### Compatibility of two definitions of elliptic elements in GLn

For an element $g$ of a connected reductive group $G$ (over a local field),
$g$ is called $elliptic$ if it is semisimple and the maximal split subtorus of the center of the centralizer of $g$ is ...

**2**

votes

**1**answer

112 views

### Is any/every order on a number field forced by some finite extension?

Say a field extension $E/F$ forces the order on an ordered $F$ if every positive $x$ in $F$ is a sum of squares in $E$. A real closure of $F$ does this. And $\mathbb{Q}$ forces its own sole ...