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9
votes
2answers
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
1answer
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
2answers
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
1answer
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
1answer
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
1answer
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
5answers
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
3answers
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
1answer
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
1answer
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
votes
4answers
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
2answers
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
1answer
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
1answer
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
2answers
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
1answer
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
2answers
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
4answers
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
3answers
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
votes
4answers
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
1answer
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
3answers
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
2answers
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
3answers
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
votes
3answers
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
votes
5answers
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
3answers
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
3answers
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
2answers
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
votes
5answers
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
4answers
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
2answers
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
votes
2answers
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
1answer
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
1answer
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
8answers
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
6answers
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
5answers
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 ...