The number-fields tag has no usage guidance.

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**1**answer

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### Number of representations as sums of squares in rings of integers of number fields

Let $K$ be some number field, $\mathcal O_K$ denote its ring of integers, and let $n$ be a positive integer. Take $\alpha \in \mathcal O_K$, and consider the quantity $r_{n,K}(\alpha)$, which denotes ...

**1**

vote

**1**answer

91 views

### Freeness of the group of principal ideals of a number field

This is just a question wondering whether a concrete (enough) result that can be proved using the axiom of choice can be proved without it. The result being that the group of principal ideals $P_K$ of ...

**3**

votes

**1**answer

163 views

### A question about “small” sets of algebraic numbers

For the sake of brevity let me use the following terms. A subset $X$ of $\bar{\mathbb{Q}}$ will be called "small" if for any number field $K$, the intersection
$X\cap K$ is finite. Similarly, a set ...

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votes

**0**answers

153 views

### Normal basis in cyclotomic number fields

Let $p$ be an odd prime integer and let $\zeta$ be a primitive $2p$-th root of unity. Does $\alpha=1+\zeta+\zeta^{-1}+\dots+\zeta^{\frac{p-1}{2}}+\zeta^{-\frac{p-1}{2}}$ generates a normal basis of ...

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votes

**1**answer

148 views

### “frequency” of fields for which the p-adic regulator vanishes (mod p)

There is a very nice question which arises in the study of the
Discrete Logarithm Problem which I wish to present here.
The question, in a general setting, is to specify an empirical
expression for ...

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votes

**1**answer

166 views

### Normal basis with cyclotomic units

Let p be an odd prime integer and let $\zeta$ be a primitive p-th root of unity.
Let $\alpha$ be a non-trivial cyclotomic unit of $\mathbb Q(\zeta)$, i.e. an element of the form ...

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votes

**1**answer

390 views

### Do all algebraic number fields arise from Eisenstein polynomials?

This question came up while going through the application of Eisenstein criterion: The $p$-th cyclotomic polynomial after changing the variable $x$ to $(x+1)$ satisfies Eisenstein criterion. That is ...

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**2**answers

571 views

### Isomorphism problem for two radical extensions

Let $n\geq 2$ and let $a,b\in{\mathbb Q}$. Suppose that both the
polynomials $A=X^n-a$ and $B=X^n-b$ are irreducible. We want to know whether
( * ) there is a root $\alpha$ of $A$ and a root $\beta$ ...

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**0**answers

100 views

### construct totally real cubic fields

Given some $e_i=0$ or $1$ for $1\le i \le 3$.
I wish to construct a totally real cubic number field $K$ so that $\prod_{i=1}^3 sgn(\sigma_i(u))^{e_i}$ is always $1$ for any $u\in O_K^\times$ where ...

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**0**answers

131 views

### Siegel Walfisz Theorem for algebraic number fields

Is there a generalization of the Siegel Walfisz to algebraic number fields? This has been done for the prime number theorem in the prime ideal theorem.

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111 views

### Lowest degree polynomial with integer coefficients yielding $1/\sqrt{2^n}$

Let $x = \cos(\pi/8) = \frac{1}{2} \sqrt{2+\sqrt{2}}$ and $y = \sin(\pi/8) = \frac{1}{2} \sqrt{2-\sqrt{2}}$. What is the lowest degree polynomial $p(x,y)$ with integer coefficients such that $p(x,y) = ...

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**1**answer

327 views

### Are the Szpiro ratios of 37b1 over certain number fields {33,39,42,48,51,66}?

Related to this question.
According to Hindry p.7 Conj 3.1
and Stein's notes Szpiro's conjecture over number fields states that the Szpiro ratio is:
$$ ...

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votes

**1**answer

1k views

### Go I Know Not Whither and Fetch I Know Not What

Next day: apparently my original question is harder, by far, than the other bits. So: it is a finite check, I was able to confirm by computer that, if the polynomial below satisfies $$ f(a,b,c,d) ...

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vote

**0**answers

170 views

### Is it trivial to get $200$ algebraic abc triples of equal quality $1.6978…$ over isomorphic number fields?

Got $200$ algebraic abc triples over distinct though isomorphic
number fields of equal quality $1.6978...$
Strongly suspect I can get as many as I like
(assuming the computations are correct).
Is ...

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votes

**2**answers

545 views

### Can there be a power basis for a totally real field of high degree?

A number field $K$ is said to have a power basis if there is an $\alpha \in K$ such that the full ring of integers $O_K$ is the $\mathbb{Z}$-linear span of ...

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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 ...

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**4**answers

3k 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 ...

**5**

votes

**1**answer

254 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 ...

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votes

**9**answers

6k 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
$\mathbb{C}$,
$\mathbb{R}$, ...

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votes

**1**answer

369 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 ...

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votes

**2**answers

226 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 ...

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**1**answer

317 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 ...

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**0**answers

1k 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 ...

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**1**answer

205 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 ...

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**2**answers

390 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 ...

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vote

**1**answer

295 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 ...

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**1**answer

78 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 ...

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**1**answer

133 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 ...

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votes

**2**answers

361 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 ...

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**1**answer

129 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 ...

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votes

**2**answers

354 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 ...

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**2**answers

401 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 ...

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votes

**1**answer

112 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 ...

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vote

**2**answers

575 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

**1**answer

572 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 ...

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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 ...

**3**

votes

**1**answer

139 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

**1**answer

204 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 ...

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votes

**0**answers

81 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 ...

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votes

**2**answers

529 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

162 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 ...

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votes

**2**answers

858 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 ...

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**0**answers

141 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 ...

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**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, ...

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**1**answer

141 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 ...

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votes

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968 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
...

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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.

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votes

**3**answers

378 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 ...

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votes

**2**answers

617 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 ...

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**0**answers

120 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, ...