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### Primitive element for a number field, and ramification

Let $K=\mathbb Q(\theta)$ be a number field with integral primitive element $\theta$, and let $f(x)$ be the minimal polynomial of $\theta$. Let $p$ be a rational prime. It's well known that if $p$ ...

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227 views

### Does every ring of integers sit inside a monogenic ring of integers?

Given a number field $K/\mathbf{Q}$ whose ring of integers $\mathcal{O}_K$ is, in general, not of the form $\mathbf{Z}[\alpha]$ (not monogenic), does there exist an extension $L/K$ which has ...

**2**

votes

**1**answer

96 views

### Can there be a numerical system in which logarithms can be expressed in terms of exponentials in closed form?

The invention of complex numbers allowed to express trigonometric functions through hyperbolic ones in closed form.
Is there possible an extension of real/complex numbers in which logarithms and ...

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104 views

### More generalized RSA construction

Is there a way to construct RSA type cryptosystem over general number rings?
Can Number Field Sieve technique be applied here?

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**0**answers

182 views

### Automorphism group of $\mathbf{C}$ over $\overline{\mathbf{Q}}$

Choose an embedding $\overline{\mathbf{Q}}\rightarrow \mathbf{C}$ from an algebraic closure of the field of rationals to the field of complex numbers.
Question 1: Is it true that $\mathbf{C}$ is ...

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**2**answers

838 views

### Can the Dedekind zeta function distinguish between real and imaginary quadratic number fields?

Suppose I am given a machine that gives me the coefficients $a_1$, $a_2$, $a_3$, ... of a Dirichlet series
$$\sum_1^{\infty} \frac{a_n}{n^s} $$
and assume that I know that this Dirichlet series is the ...

**7**

votes

**1**answer

506 views

### Given $v,w$ primes of $k$, is there $K/k$ so $K_v\cap\Bbb Q^{cycl}=K_w\cap\Bbb Q^{cycl}=K\cap\Bbb Q^{cycl}$?

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 ...

**6**

votes

**1**answer

153 views

### Unramified extensions of quadratic fields

Let $K/\mathbb{Q}$ be quadratic and let $L/K$ be an (everywhere) unramified Galois extension. If $L/K$ is abelian, then one can show that $L/\mathbb{Q}$ is Galois (eg see here). Is $L/\mathbb{Q}$ ...

**15**

votes

**1**answer

661 views

### What's so special about these $17$th deg equations?

While browsing the Database of Number Fields, I came across 17T8. It only had four equations, one of which is,
$$\small{x^{17} - 5x^{16} + 40x^{15} - 140x^{14} + 610x^{13} - 1622x^{12} + 4870x^{11} - ...

**15**

votes

**1**answer

824 views

### A set of generators for $\bar{\mathbb{Q}}$

Two questions:
Does there exist a sequence $\alpha_1,\alpha_2,...$ of algebraic numbers with degrees $d_1,d_2,...$ s.t. for each $i$, $d_i|d_{i+1}$ and $\alpha_i= p_i(\alpha_{i+1})$ with $p_i$ a ...

**32**

votes

**3**answers

977 views

### Simple argument regarding sums of two units in a number field?

I wonder if it is possible to show, without using the Schmidt subspace/Roth theorem/Baker's bounds on linear forms in logarithms or other very deep results, that, in a number field, not all integral ...

**15**

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, ...

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votes

**2**answers

236 views

### 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 ...

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**0**answers

156 views

### Writing integers in ring of integers of number fields

Given $a,b\in\Bbb N$, we can write $a=a_tb^t+a_{t-1}b^{t-1}+\dots+a_1b+a_0$ where $t=\lceil\log_ba\rceil$ and $a_i<b<a$.
(1) Supposing if $b\in\mathcal{O}_K$ where $\mathcal{O}_K$ is ring of ...

**1**

vote

**1**answer

97 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

173 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 ...

**2**

votes

**0**answers

165 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 ...

**2**

votes

**1**answer

158 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 ...

**0**

votes

**1**answer

176 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 ...

**6**

votes

**1**answer

417 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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votes

**2**answers

587 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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117 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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148 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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116 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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votes

**1**answer

345 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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186 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

654 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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3k 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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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 ...

**7**

votes

**1**answer

270 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

**9**answers

7k 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

**1**answer

389 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

320 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 ...

**7**

votes

**1**answer

361 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 ...

**23**

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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 ...

**1**

vote

**1**answer

235 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**

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397 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 ...

**3**

votes

**1**answer

303 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

98 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

140 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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**2**answers

365 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

132 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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372 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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472 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**

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**1**answer

114 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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**2**answers

657 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 ...

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votes

**1**answer

582 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

4k 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 ...

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**1**answer

146 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 ...