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431 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

643 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

462 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

249 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 $k$ ...

**18**

votes

**1**answer

665 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, ...

**4**

votes

**5**answers

1k 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

472 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

875 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

421 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 ...

**7**

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**4**answers

1k 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

481 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

251 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

851 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

617 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 :
...

**6**

votes

**1**answer

455 views

### Roots of unity in different completions of a number field

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

775 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

3k 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

555 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 ...

**22**

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**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 ...

**6**

votes

**1**answer

598 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

293 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

745 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**

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**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 ...

**13**

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**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.

**14**

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

1k 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

365 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 ...

**12**

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**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 ...

**10**

votes

**4**answers

2k 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

352 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**

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**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

608 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

768 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

2k 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 ...