The number-fields tag has no usage guidance.

**4**

votes

**2**answers

348 views

### Subject to some conditions, is it possible to conclude a subfield of an abelian extension generated by a unit is a cyclic extension

My research is mostly in the area of modular categories. In the course of my research I came across a constraining set of number theoretic conditions that I'd like to exploit. It has been pointed out ...

**0**

votes

**0**answers

173 views

### Ring of Integers as subring with most irreducibles

Let $L$ be a number field. Is it possible to define its ring of integers $R$ by saying it's the subring with (in a fuzzy sense) the "most" irreducibles?

**5**

votes

**0**answers

489 views

### Elliptic curve with no points in a number field

The following is probably well-known (I'd appreciate a link):
for a field $K$ that is a finite extension of the field of rational numbers, give a polynomial $f(x,y) ∈ Q[x,y]$ of the form $y^2 − x^3 ...

**2**

votes

**0**answers

171 views

### bounds for the covering radius (or diameter of the Voronoi cell) of a lattice coming from a number field

Let $K$ be a number field and $O$ an order in $K$. Denote the real embeddings of $K$ by $\sigma_1,\dots,\sigma_s$ and the non-real complex embeddings of $K$ by ...

**11**

votes

**2**answers

363 views

### Records for low-height points on elliptic curves over number fields

Elkies maintains a list of nontorsion points of low height on elliptic curves over Q; does anyone know of anything similar for curves over number fields?
Everest and Ward give examples of points of ...

**14**

votes

**4**answers

2k views

### $Q(\sqrt{2})=Q((\sqrt{2}+1)^n)$

Observe that we have $Q(\sqrt{2})=Q((\sqrt{2}+1)^n)$.
More generally, assume that $K$ is a finite extension of Q. Is there any $\alpha \in K$ such that $K=Q(\alpha^n)$ for every $n \in N$?

**4**

votes

**1**answer

636 views

### Is there a section disjoint from 0, 1 and infinity on the projective line

Let $K$ be a number field with ring of integers $O_K$. Is there a section of $\mathbf{P}^1_{O_K}$ over $O_K$ whose image is disjoint from $0$, $1$ and $\infty$? If $K=\mathbf{Q}$ this is not possible ...

**1**

vote

**1**answer

328 views

### The union of the totally split primes

Let $R$ be a Dedekind domain with quotient field $K$, let $L$ be a finite separable extension of $K$, and let $S$ be the integral closure of $R$ in $L$. If $\mathfrak{p}$ is a nonzero prime ideal of ...

**20**

votes

**2**answers

1k views

### The divisor bound in number fields

The divisor bound asserts that for a large (rational) integer $n \in {\bf Z}$, the number of divisors of $n$ is at most $n^{o(1)}$ as $n \to \infty$. It is not difficult to prove this bound using the ...

**10**

votes

**1**answer

606 views

### local-global principle for units

Say that $L/K$ is a quadratic extension of number fields with $K$ totally real and $L$ totally imaginary.
Then the Hasse norm theorem says that an element of $K$ that is everywhere a local norm is ...

**2**

votes

**2**answers

622 views

### Which numbers appear as discriminants of cubics?

I'm trying to show that all possible splitting fields occur for a class of cubic polynomials, so have started by looking at the discriminants.
Clearly, given that they are products of squares, all ...

**13**

votes

**2**answers

473 views

### Which polynomials arise as formulas for a conjugate

For any integer $r \geq 2$, et $V_r$ be the set of polynomials $Q \in {\mathbb Q}[X]$ of degree $r-1$ such that there is an algebraic number $\alpha$ of degree $r$ , such that
$Q(\alpha)$ is a ...

**2**

votes

**1**answer

654 views

### A subring question (revised)

Hello,
Let $K/{\mathbb Q}$ be a finite extension which is not necessarily Galois, and ${\mathcal O}$ be the ring of integers of $K$. Let $p$ be a prime in ${\mathbb Q}$ and let
$p {\mathcal ...

**8**

votes

**2**answers

467 views

### If the n-th root of unity exists locally, does it exist globally?

My question is: is it true that $\zeta_n$ is in $K$ (a number field) iff $\zeta_n$ is in all but finitely many of the $K_{\mathfrak{p}}$?

**18**

votes

**1**answer

1k views

### Are class numbers encoded in the absolute Galois group of ${\mathbb Q}$?

The absolute Galois group $G_{\mathbb Q}=\text{Gal}(\bar{\mathbb Q}/\mathbb Q)$, as a profinite group, encodes a lot of things: the whole lattice of number fields (closed subgroups of finite index), ...

**0**

votes

**1**answer

251 views

### extensions of number fields

Let $k$ be a number field, and $F/k$ a finite extension. I would like to find a countable family of extensions $k_i/k$ of degree 2 and a place $v_i$ of $k_i$ such that if $v$ is the place of $k$ ...

**19**

votes

**1**answer

715 views

### How random are unit lattices in number fields?

I was wondering how random unit lattices in number fields are. To make this more precise:
If $K$ is a number field with embeddings $\sigma_1, \dots, \sigma_n, \overline{\sigma_{r+1}}, \dots, ...

**5**

votes

**5**answers

2k views

### Why is every quadratic subfield of a Galois extension of the rationals with the quaternions as Galois group real?

Suppose that L is a field extension of the rationals with Galois group the quaternions Q={1,-1,i,-i,j,-j,k,-k}. Furthermore assume that L contains a quadratic subfield K. I have learned from this link ...

**1**

vote

**3**answers

502 views

### Given an integer n and a finite extension K of Q , find a polynomial of degree n that is irreducible over K

Given a positive integer n and a finite extension $K$ of $\mathbb{Q}$, can one always find an irreducible polynomial in $K[x]$ of degree n? What if $n$ is prime?
The natural approach is to take a ...

**18**

votes

**1**answer

936 views

### What is the ring of integers of the Pythagorean field?

Following Hilbert, we call the complex numbers constructible via
compass and straight-edge the field of Euclidean numbers, and
the totally real such numbers the field of Pythagorean numbers. (Among ...

**1**

vote

**1**answer

504 views

### Number of subset sums

Let $\mathbb{F}_q$ be a finite field with characteristic $p$ and $p < q$ (i.e. not a prime field). Let $D\subseteq \mathbb{F}_q$ be a some set with $|D|=n$. Find a non-empty subset ...

**8**

votes

**4**answers

2k views

### Extension of valuation

Fix a prime number $p$. Suppose that I have a valuation $v_p: \mathbb{Q} \to \mathbb{Q}$ on the rationals $\mathbb{Q}$. That is, $v_p( p^n(\frac{a}{b})) = p^{-n}$ where each of $a,b$ is not divisible ...

**6**

votes

**2**answers

491 views

### What is the set of possible values of the degree of the sum of two algebraic numbers with fixed degrees?

This question is related to Degree of sum of algebraic numbers and algebraic numbers of degree 3 and 6, whose sum has degree 12.
In this last question I asked a very special case of the following ...

**3**

votes

**1**answer

257 views

### Distribution of associate primes modulo q in number fields

In Dirichlet's theorem for number fields, I asked about an analogue of Dirichlet's theorem (or I guess I should call it the Prime Number Theorem for Arithmetic Progressions) for number fields. ...

**5**

votes

**1**answer

1k views

### Dirichlet's theorem for number fields

I'd like to see a formulation of Dirichlet's theorem for number fields, i.e. some analogue of the assertion:
The number of primes less than $N$ congruent to $a \pmod{m}$ where $(a,m)=1$ is
...

**9**

votes

**2**answers

635 views

### Why the roots of unity are the analogs of constants ?

Hello,
Joel Dogde, in a comment on his question "Roots of unity in different completions of a number field", says the following, about the analogy between number fields and function fields :
...

**7**

votes

**1**answer

506 views

### Given $v,w$ primes of $k$, is there $K/k$ so $K_v\cap\Bbb Q^{cycl}=K_w\cap\Bbb Q^{cycl}=K\cap\Bbb Q^{cycl}$?

For any field $k$, let $\mu(k)$ denote the roots of unity in $k$. Now let $k$ be a number field and let $v, w$ be non-archimedean primes of $k$ with distinct residual characteristics. Does there ...

**7**

votes

**2**answers

855 views

### number fields generated by units of number fields

Which number fields are generated by the units of some number field? That is, if $K$ is a number field and $U(K)$ its group of units, the field $k = \mathbb{Q}(U(K))$ is a subfield of $K$. But which ...

**42**

votes

**4**answers

4k views

### Degree of sum of algebraic numbers

This is an elementary question (coming from an undergraduate student) about algebraic numbers, to which I don't have a complete answer.
Let $a$ and $b$ be algebraic numbers, with respective degrees ...

**5**

votes

**3**answers

603 views

### Transformations of integer polynomials under combinations of their roots

I'm wondering whether the following ideas/questions give rise to an existing body of research. (Accordingly: please suggest appropriate changes to the tags!)
Preamble
We consider polynomials ...

**23**

votes

**4**answers

2k views

### Which number fields are monogenic? and related questions

A number field $K$ is said to be monogenic when $\mathcal{O}_K=\mathbb{Z}[\alpha]$ for some $\alpha\in\mathcal{O}_K$. What is currently known about which $K$ are monogenic? Which are not? From ...

**8**

votes

**1**answer

660 views

### Is there an analog of class field theory over an arbitrary infinite field of algebraic numbers?

Recently, I found a paper by Schilling http://www.jstor.org/pss/2371426, which mentions that for certain infinite field of algebraic numbers there is an analog of class field theory. By infinite field ...

**3**

votes

**3**answers

1k views

### Why isn't there a structure with two primes?

I don't know whether this question is a bit too vague for MO or not, so feel free to delete it if you see fit.
The p-adic integer is defined by taking the inverse limit $\ldots \mathbb{Z} / p^2 ...

**1**

vote

**2**answers

303 views

### Infinite collection of elements of a number field with very similar annihilating polynomials

Hello all, let $n$ be an integer $\geq 2$ and let $\alpha$ be an algebraic number
of degree $n$. Let $R$ be the ring of algebraic integers in ${\mathbb Q}(\alpha)$, and
let $B$ be the subset of $R$ ...

**2**

votes

**3**answers

778 views

### Unit in a number field with same absolute value at a real and a complex place

I was asked whether it was possible to produce a monic polynomial with integer coefficients, constant coefficient equal to $1$, having a real root $r > 1$ and a pair of complex roots with absolute ...

**12**

votes

**3**answers

2k views

### Computing (on a computer) higher ramification groups and/or conductors of representations.

I am supervising an undergraduate for a project in which he's going to talk about the relationship between Galois representations and modular forms. We decided we'd figure out a few examples of weight ...

**14**

votes

**5**answers

2k views

### Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$?

Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$? I am going to be on the road soon, so pleas don't be offended if I don't respond quickly to a comment.

**15**

votes

**3**answers

2k views

### sum of squares in ring of integers

Lagrange proved that every (positive) rational integer is a sum of 4 squares.
Are there general results like this for ring of integers of a number field? Is this class field theory?
Explicity, ...

**4**

votes

**3**answers

2k views

### How many primes stay inert in a finite (non-cyclic) extension of number fields?

In the following suppose L/K is a finite Galois extension of number fields, (maybe it works for other cases also, I don't know) By the Chebotorev density theorem when Gal(L/K) is cyclic, there are ...

**2**

votes

**2**answers

394 views

### Continuation up to zero of a Dirichlet series with bounded coefficients

Let $a_n$ be a bounded sequence of positive real numbers. Is it the case that the Dirichlet series $\sum \frac{a_n}{n^s}$ can be meromorphically continued up to the right of zero, with at the most a ...

**14**

votes

**5**answers

1k views

### Families of number fields of prime discriminant

When I am testing conjectures I have about number fields, I usually want to control the ramification, especially minimize to a single prime with tame ramification. Hence, I usually look for fields of ...

**11**

votes

**4**answers

3k views

### A problem on Algebraic Number Theory, Norm of Ideals

A problem on Algebraic Number Theory
$K$ and $L$ are number fields over $\Bbb Q$.($\Bbb Q$ is rational number field)
$K\subseteq L$.
$\mathcal{O}_K$ is the ring of integers of $K$.
and ...

**9**

votes

**2**answers

357 views

### Subfields joining an algebraic element to another

Let $\alpha$ and $\beta$ be two algebraic numbers over $\mathbb Q$. Say that a subfield $\mathbb K$ of $\mathbb C$ joins $\alpha$ to $\beta$ iff $\beta \in {\mathbb K}[\alpha]$ but $\beta \not\in ...

**10**

votes

**2**answers

1k views

### Irreducible polynomial over number field with roots in every completion?

Let K/Q be a field, probably not a finite extension. Is it possible for a polynomial to be irreducible over K but have a root in every completion of K? What about all but finitely many completions?
...

**0**

votes

**1**answer

621 views

### On algebraic field extensions

Let $L:K$ be a field extension. Let $A$ be a set of elements in $L$, all of which are algebraic over $K$. Construct the field extension $M=K(A)$. I have two questions:
[1] Is $M:K$ an algebraic field ...

**1**

vote

**1**answer

816 views

### Neukirch's class field axiom and cohomology of units for unramified extension

This question may be too detailed but perhaps somebody knows the answer: Neukirch proofs in his algebraic number theory book in chapter IV, proposition 6.2, that his class field axiom implies that the ...

**9**

votes

**8**answers

3k views

### Which computer algebra system should I be using to solve large systems of sparse linear equations over a number field?

This is related to Noah's recent question about solving quadratics in a number field, but about an even earlier and easier step.
Suppose I have a huge system of linear equations, say ~10^6 equations ...

**14**

votes

**6**answers

1k views

### Solving polynomial equations when you know in which number field the solutions live

Suppose I have a bunch of polynomial equations with coefficients in a number field, and suppose further that I'm guaranteed a priori that they have a solution in that number field. Can I leverage ...

**22**

votes

**5**answers

2k views

### Global fields: What exactly is the analogy between number fields and function fields?

Note: This comes up as a byproduct of Qiaochu's question "What are examples of good toy models in mathematics?"
There seems to be a general philosophy that problems over function fields are easier to ...