The number-fields tag has no wiki summary.
0
votes
1answer
254 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
1answer
63 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 ...
0
votes
0answers
14 views
sufficiency of the relative discriminant to be a square of an ideal for an unramified quadratic extension
Is it sufficient for a quadratic extension of a cubic number field to have a relative discriminant as a square of an ideal for being unramified extension (excluding primes dividing 2 for the sake of ...
0
votes
1answer
101 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 ...
0
votes
0answers
36 views
existence of elements with specific norms in pure cubic fields
Is there any specific way to find an element with a given norm in pure cubic field? say for an example an element of norm 5 in pure (monogenic) cubic field of 11.
it is easy to check that 5 as an ...
3
votes
1answer
123 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
2answers
303 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
2answers
259 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
1answer
101 views
One parameter families of elliptic curves over rings of integers of number fields
Let $A(n), B(n) \in \mathbb{Z}[n]$ be polynomials, not both constant, such that $4A^3(n) + 27B^2(n)$ is not the zero polynomial and the polynomial (in variables $x, y$) $$y^2 - x^3 - A(n)x - B(n) \in ...
2
votes
1answer
116 views
Irreducibility of trinomials over number fields
I wonder if the following is known or, not very difficult to see:
Let $K$ be a number field and $\alpha \in K$ be nonzero. Does there necessarily exist a positive integer $n > 1$ such that the ...
3
votes
1answer
135 views
Irreducibility of trinomials
I wonder if the following is known or, not very difficult to see:
Let $K$ be a number field and $A, B \in \mathcal{O}_K$ be nonzero integers of $K$. Does there necessarily exist a positive integer $n ...
0
votes
0answers
68 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
2answers
515 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 $K$ ...
3
votes
0answers
136 views
Irreducibility of the trinomial over Q
I'm trying to find an algebraic proof of irreducibility of the polynomial $x^n-x-1$ over rational numbers (or integers, which the same). I've read the Selmer's paper "On the irreducibility of certain ...
1
vote
1answer
149 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 ...
7
votes
2answers
413 views
Quintic polynomial solution by Jacobi Theta function.
Does someone have a good and rigorous reference for the solution of quintic ploynomial equation with Jacobi Theta function, in English?
Mathworld and Wikipedia don't give a good English reference, at ...
1
vote
0answers
124 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 ...
17
votes
0answers
776 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
1answer
128 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 ...
8
votes
4answers
875 views
The “interplay” between additive and multiplicative structure in a field
A field is an ordered triple $(F, +,\cdot)$ of a set $F$ and binary operations $+,\times$ on $F$ such that $(F,+)$ and $(F\backslash 0,\times)$ are abelian groups satisfying the distributive laws
...
2
votes
3answers
310 views
Useful notion of unramified Galois representation
Let $\mathbf C(t)$ be the field of rational functions and let $\overline{\mathbf C(t)}$ be an algebraic closure. Let $G$ be the Galois group of $\overline {\mathbf C(t)}$ over $\mathbf C(t)$.
Let ...
6
votes
2answers
541 views
Is it known if the absolute Galois group is “divisible”?
The definitions of a divisible group that I have seen all seem to assume abelian is an a priori property of the group. My question is as to whether or not it is known that--given a non-torsion element ...
2
votes
0answers
106 views
Comparing ideal class numbers of different orders
Let $P$ be a monic irreducible integral polynomial. Let $K=\mathbf Q[X]/(P)$ be the associated number field, $\mathcal O$ be its ring of integers and $R$ be the order $\mathbf Z[X]/(P)$.
(In general, ...
0
votes
1answer
539 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
0answers
207 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
1answer
245 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
2answers
334 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
0answers
163 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
0answers
447 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 ...
1
vote
0answers
130 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 ...
10
votes
2answers
333 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
4answers
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
1answer
624 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
1answer
284 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
2answers
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
1answer
540 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
2answers
441 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
2answers
418 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
635 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
456 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
241 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
1answer
617 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
462 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
833 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
389 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
476 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
247 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. ...
