# Tagged Questions

**1**

vote

**1**answer

165 views

### lattice in number field already a fractional ideal?

Let $K=\mathbb{Q}[\alpha]$ where $\alpha$ is integral over $\mathbb{Z}$ such that the Galois hull of $K$ can be embedded in $\mathbb{R}$. Let $S=\mathbb{Z}[\alpha]$. Let $x_1, \ldots , x_n$ be a ...

**7**

votes

**2**answers

377 views

### Is realness of number fields exponentially bounded?

In a given non-real algebraic number field (say, given by an irreducible polynomial over $\mathbb{Q}$) is there a complexity bound on the summands $x,\dots,t$ that make $-1$ a sum of squares? So ...

**6**

votes

**1**answer

237 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 ...

**1**

vote

**1**answer

273 views

### Number field of degree 5

I am interested in field extensions of the rationals. About degree 3 extensions there are many refrences including the famous paper of Shanks "The Simplest Cubic Fields". In particular he gave ...

**0**

votes

**1**answer

116 views

### Aurifeuillean factorization with number fields

Basically the question is if number fields can be used
in Aurifeuillean factorization.
Probably this is easy and the answer is "no".
Let $f,g \in \mathbb{Z}[x], a \in \mathbb{N}$.
Let $f(x)$ and ...

**3**

votes

**1**answer

126 views

### Definability of orderings on a formally real number field

For vector basis $b_1,..,b_n$ on a finite extension $F$ of $\mathbb{Q}$, where $-1$ is not a sum of squares, each linear order on $F$ is determined by an order on the basis. This uses information ...

**4**

votes

**2**answers

323 views

### On class numbers $h(-d)$ and the diophantine equation $x^2+dy^2 = 2^{2+h(-d)}$

Given fundamental discriminant $d \equiv -1 \bmod 8$ such that the quadratic imaginary number field $\mathbb{Q}(\sqrt{-d})$ has odd class number $h(-d)$. Is it true that one can always solve the ...

**7**

votes

**2**answers

306 views

### Number of ways to write an integer as a product of irreducibles

Is there any way to tell the number of distinct ways to factor $a\in\mathcal{O}_k$ (up to units, of course) when $k$ is not a PID? A simple investigation in $\mathbb{Q}(\sqrt{-5})$ with integer ring ...

**0**

votes

**0**answers

74 views

### Divisor bounds of ideals in number fields

Let $K$ be an algebraic number field and let $I$ be an ideal in $O_K$ (the ring of integers).
Denote by $d(I)$ the number of ideals that divide $I$.
So if $I= \prod_{i=1}^k p_i^{e_i}$ is the ...

**6**

votes

**2**answers

518 views

### Simultaneous Powers Far From 1

I'm looking for a reference or proof of the following. Let $K/\mathbb{Q}$ be a finite Galois extension of degree $n$. Let $a_1,\ldots,a_n$ be Galois conjugate elements in the ring of integers of $K$ ...

**1**

vote

**1**answer

157 views

### Lower Degree Elements in an Algebraic Number Field

Fix an algebraic integer $\alpha$ of degree $n$
such that the extension $K=\mathbf{Q}(\alpha)/\mathbf{Q}$ has intermediate fields.
(We can assume $K$ is Galois with non-simple Galois group.)
This ...

**1**

vote

**0**answers

127 views

### Full $n$-torsion of elliptic curves and the cyclotomic field of order $n$

Hi, overflowers.
I have a question concerning the torsion of elliptic curves over number fields.
Let us consider an elliptic curve $E$ defined over ${\mathbb Q}$. From the Weil pairing one can ...

**20**

votes

**0**answers

914 views

### Orders in number fields

Let $K$ be a degree $n$ extension of ${\mathbb Q}$ with ring of integers $R$. An order in $K$ is a subring with identity of $R$ which is a ${\mathbb Z}$-module of rank $n$.
Question: Let $p$ be an ...

**2**

votes

**1**answer

130 views

### ramification of discrete valuation field

Let $K$ be a discrete valuation field with valuation $v:K\rightarrow \mathbb Z\cup \{\infty\}$ which is normalized by $v(\pi)=1$ for a prime element $\pi$. Let $v:\overline K\rightarrow \mathbb ...

**0**

votes

**1**answer

552 views

### A question on Cebotarev's density theorem

Let $K$ be a number field, $d$ a positive integer and $S$ a finite set of places of $K$.
By Cebotarev, there exists a finite set of finite places $T$ disjoint from $S$ such that the conjugacy classes ...

**1**

vote

**0**answers

216 views

### The field $\mathbb{Q}(\cos \frac {2\pi} {n})$ [closed]

Let $x$ be $\cos \displaystyle \frac {2\pi} {n}$ for some natural number $n$.
Then is there an integer $n$ such that $\mathbb{Q}(x^2+x)\neq \mathbb{Q}(x)$?
I also would like to know if there is some ...

**2**

votes

**1**answer

254 views

### Is there a Dirichlet Unitary Unit Theorem?

Dirichlet's unit theorem computes the group of units of the algebraic numbers of a number field. There are a few generalisations for orders available.
Assume the order has an involution. For example, ...

**4**

votes

**2**answers

337 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 ...

**11**

votes

**2**answers

340 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 $n \in N$?

**1**

vote

**1**answer

296 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 ...

**19**

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

553 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 ...

**0**

votes

**2**answers

478 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 ...

**9**

votes

**2**answers

422 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

639 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

459 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

246 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 $k$ ...

**17**

votes

**1**answer

644 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 ...

**5**

votes

**1**answer

819 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

613 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

449 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

749 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 ...

**18**

votes

**4**answers

1k 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

581 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

291 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

734 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 ...

**13**

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.

**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 ...

**12**

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 ...

**9**

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 Q.(Q is rational number filed)
K is a subfield of L.
O_K is the integers of K.
and O_L is the integers of L.
P is a prime ideal ...

**9**

votes

**2**answers

349 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 ...

**9**

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?
...

**1**

vote

**1**answer

761 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 ...