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4
votes
1answer
189 views

Continued fraction expansion of an algebraic number and its conjugates

Let $w$ be an element of a Galois extension $L:\mathbb{Q}$ such that $\text{Gal}(L/\mathbb{Q})=\langle g\rangle$ is cyclic of order $n$ (here $\mathbb{Q}$ is rationals). Suppose we know the continued ...
23
votes
9answers
5k views

Why are polynomials so useful in mathematics?

This is perhaps unanswerable, or perhaps I am too algebraically ignorant to phrase it cogently, but: Is there some identifiable reason that polynomials over $\mathbb{C}$, $\mathbb{R}$, ...
9
votes
1answer
340 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 ...
4
votes
2answers
125 views

unit group of biquadratic fields

In the unit group of a real biquadratic field, what is the index of the product of the unit groups of its three quadratic subfields? Is the index 1 if discriminant of these three subfields are always ...
18
votes
4answers
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
1answer
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 ...
20
votes
0answers
912 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 ...
1
vote
1answer
164 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
2answers
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 ...
1
vote
1answer
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
1answer
69 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
33 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
115 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 ...
11
votes
2answers
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 ...
0
votes
0answers
54 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
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
2answers
322 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
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
1answer
105 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 ...
0
votes
2answers
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 ...
0
votes
1answer
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 ...
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 ...
3
votes
1answer
128 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
158 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
73 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
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
1answer
156 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
504 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
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 ...
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, ...
2
votes
1answer
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 ...
8
votes
4answers
900 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 ...
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.
2
votes
3answers
333 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
555 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
110 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, ...
1
vote
0answers
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 ...
9
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 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 ...
2
votes
1answer
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
2answers
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 ...
0
votes
0answers
167 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?
1
vote
1answer
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 ...
5
votes
0answers
457 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
135 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 ...
4
votes
1answer
628 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 ...
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 $n \in N$?
1
vote
1answer
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 ...
10
votes
1answer
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 ...
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 ...
17
votes
1answer
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, ...