# Questions tagged [number-fields]

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**9**

votes

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144 views

### How small may the discriminant of an $S_d$-field be?

In every degree $d$, the Galois closure of the typical number field has the maximal possible Galois group $S_d$. Denote by $f(d)$ the least absolute value of a discriminant of an $S_d$-field of degree ...

**2**

votes

**1**answer

100 views

### why the division field of an abelian variety contains a cyclotomic field?

Given an abelian variety $A$ defined over $\mathbb{Q}$, for a positive integer (we can suppose prime) $\ell$, let $A[\ell]$ denote the group
of points of $A$ that are annihilated by $\ell$, the ...

**11**

votes

**1**answer

293 views

### Poisson summation formula for number fields

Poisson summation formula is widely used in many parts of the litterature, its classical formulation for sums over integers as well as its adelic version. What is its corresponding form for more ...

**4**

votes

**0**answers

73 views

### Some Questions regarding the fine Shafarevich-Tate group

The fine Shafarevich-Tate (ST) group appears to not have been studied in much depth except for one paper by Wuthrich (2007). However, it only talks about fine ST group for elliptic curves, which makes ...

**2**

votes

**0**answers

44 views

### Is there a result that connects the discriminant of an order to the discriminants of the generators of an integral basis for the order?

Let $\left\{ 1 , \omega_1, \dots , \omega_{n-1} \right\}$ be an integral basis for an order $\mathcal{O}$ of a number field of degree $n$ over $\mathbb{Q}$, let $\Delta $ be the discriminant of $\...

**17**

votes

**4**answers

495 views

### In which cyclic cubic number fields does there exist this type of unit?

Let $K$ be a cyclic cubic number field with conductor $f$ and ring of integers $\mathcal{O}_K$.
Define $K$ to be blue if and only if $$\operatorname{Norm}_{K/\mathbb{Q}}(w) = \operatorname{Norm}_{K/...

**5**

votes

**1**answer

240 views

### Explicit family of number rings $\mathcal{O}_{K_n}$ requiring $n$ generators?

Could someone provide or point me to a family of number rings $\mathcal{O}_{K_n}$ that require $n$ generators (as $\mathbb{Z}$-algebra)? Second best would be a family requiring $f(n)$ generators for a ...

**12**

votes

**1**answer

268 views

### Determine $2^{\frac{p-1}{4}}\equiv 1\pmod p$ or $2^{\frac{p-1}{4}}\equiv -1\pmod p$ when $p\equiv 1 \pmod 8$

Let $p=8k+1\equiv 1\pmod 8$ be a prime, thus $2$ is a quadratic residue module $p$. Euler's criterion show that $$2^{\frac{p-1}{2}}\equiv 1 \pmod p.$$
So we must have
$$2^{\frac{p-1}{4}}\equiv \...

**6**

votes

**0**answers

202 views

### What are the fastest ways to calculate class number of number fields?

Given a number field $K$, which approaches help us to calculate the class number $h(K)$ of $K$?
I am aware that the question is broad but any argument would be helpful.
Some basic approaches I know:...

**2**

votes

**0**answers

170 views

### Analytic class number formula for orders

In the article "The analytic class number formula for orders in products of number fields" (https://arxiv.org/pdf/1604.04564.pdf), it is shown that the analytic class number formula holds for ...

**15**

votes

**1**answer

616 views

### Are the rationals definable in any number field?

Let $K$ be a number field. Is it necessarily true that $\mathbb{Q}$ is a first-order definable subset of $K$? Equivalently (since in any number field, its ring of integers is a definable subset), is $\...

**0**

votes

**1**answer

72 views

### Field of constants of a Galois extension of function fields

Let $F$ be a finite Galois extension of the rational function field $\mathbb Q(x)$. Let $k$ be the field of constants of $F$, i.e., the algebraic closure of $\mathbb Q$ in $F$. Is $k$ necessarily a ...

**2**

votes

**2**answers

122 views

### Formulas for the structure constants of a field extension basis given by a primitive element

Let $L/K$ be a finite separable field extension and let $\theta$ be a primitive element for $L/K$ with minimal polynomial $\mu(t) \equiv \mu_{\theta/K}(t) = \sum_{k=0}^n c_k t^k$. I am trying to ...

**0**

votes

**1**answer

833 views

### What is special about 2 + $\sqrt{3}$?

Well, one thing is special about it, but it takes a while to explain.
Please let me know, whether this number occurs in other special occasions as well.
The explanation: Let $p$ be a complex ...

**6**

votes

**1**answer

264 views

### Unramified non-abelian extension and Galois cohomology

Is there an example of a finite Galois extension $E/F$ of number fields, such that $G=\mathrm{Gal}(E/F)$ is non-abelian and the order of the cohomology group $H^1(G,U_E)$ is relatively prime to class ...

**2**

votes

**1**answer

273 views

### The Genus field and Hilbert class field

Is there an example of a number field $K$ for which the genus field of $K$ is contained strictly in the Hilbert class field of $K$?

**3**

votes

**0**answers

96 views

### Interpretation or application of this analog of minimal polynomial

Recently I was thinking about images of number field elements under a polynomial with coefficients in a smaller field, and I came across the following construction. It did not have the properties I ...

**1**

vote

**1**answer

175 views

### On largest degree of polynomial related to cyclotomic polynomials - I

We know cyclotomic polynomials $\Phi_{2^kp^rq^m}(x)$ have coefficients in $\{0,\pm1\}$.
What is the largest degree $f_{d,n}(x)=\frac{x^{2^{k}p^{r}q^{m}}-1}{\Phi_d(x)}$ with $\{0,\pm1\}$ coefficients ...

**1**

vote

**1**answer

63 views

### Decomposition forms in Frobenius extensions

Let K be a Frobenius extension of Q of degree m. For a prime p, can we determine the decomposition form of p in K? Indeed, there exists a theorem due H. Cohen (Advanced topics in Computational number ...

**11**

votes

**0**answers

351 views

### A congruence involving roots of unity

Let $f(x) \in \mathbb{Z}[x]$ and suppose $f(\omega^j) \in \mathbb{Z}$ for all $j= 1, \dots, n$ where $\omega = e^{2 \pi i/n}$ is a primitive $n^{\text{th}}$ root of unity.
Computational evidence ...

**8**

votes

**0**answers

151 views

### The density of minimal polynomials

Let $K$ be a number field, i.e., a finite extension of $\mathbb{Q}$. By the primitive element theorem there exists $\alpha \in K$ such that $K = \mathbb{Q}(\alpha)$. Let
$$\displaystyle S_K = \{\...

**0**

votes

**0**answers

66 views

### How we can compute the “principal factors” in a quadratic number field?

Let K be a quadratic number field with discriminant D. It's well known that a principal factor of K exists if and only if the fundamental unit of K has norm +1. Also I know that a principal factor of ...

**15**

votes

**2**answers

642 views

### Determining the Mordell-Weil group of a universal elliptic curve

Let $K$ be a number field and let $K(a,b)$ be the field of rational functions with two indeterminates over $K$. Consider the elliptic curve $E$ over $K(a,b)$ defined by the Weierstrass equation
\begin{...

**0**

votes

**0**answers

224 views

### Extensions of $\mathbb{Z}[\sqrt{-n}]$ that are UFD

EDIT: Since the original question was to vague I will pose some stronger conditions:
Given $n \in \mathbb{N}$ with $n \geq 3$ I want to know if there is a ring $R$ with the following properties:
$\...

**8**

votes

**0**answers

142 views

### The Stickelberger's annihilation theorem over an arbitrary number field

Let $p$ be a prime and let $C = \mathbb{F}_p^\times$. Then Gal$(\mathbb{Q}(\zeta_p)/\mathbb{Q})$, where $\zeta_p$ is a primitive $p$th root of unity, may be identified with $C$ in the obvious way. Let ...

**5**

votes

**1**answer

148 views

### Are there primitive quartic CM fields whose norms of units give all totally positive units of the real quadratic subfield?

Let $K$ be a primitive (i.e. not biquadratic) quartic CM-field. That is we have $[K:\mathbb{Q}]=4$ and let $K_0=\mathbb{Q}(\sqrt{d})$ be the totally real quadratic subfield, here $d> 1$, that is we ...

**2**

votes

**1**answer

139 views

### An upper bound of number of some fractional ideals

Consider a number field $K$ of degree $n>2$, and an order $\mathcal{O}\subset \mathcal{O}_K$ on it. Let $I(\mathcal{O})$ be the monoid of $\mathcal{O}$-fractional ideals modulo principal fractional ...

**4**

votes

**1**answer

194 views

### Annihilator of a closed subgroup of adeles

Introduction:
Let $K$ be a number field and let's denote with $\mathbb A_K$ the ring of adeles which is also a locally compact group (with respect to the addition). Remember that the topology is the ...

**2**

votes

**0**answers

45 views

### Binary quadratic form which takes values that are rational time a square

Let $\mathbb{K}$ be a totally real number field, $a$ an element of $\mathbb{K}$. Consider the quadratic form $q(X,Y) = X^2-aY^2$ and assume that for every number $c$ represented by $q$, if $c$ has the ...

**1**

vote

**0**answers

171 views

### Can we choose $h(\mathcal{O})$ elliptic curves such that they have same trace and endomorphism ring $\mathcal{O}$ but distinct j-invariants?

Let $\mathbb{F}_q$ be a finite field of characteristic $p$ with $q=p^a$ elements. Let $E$ be an ordinary elliptic curve defined over $\mathbb{F}_q$. We know that $\text{End}(E)\otimes\mathbb{Q}$ is ...

**2**

votes

**0**answers

51 views

### Minimal diameter of a class in a number field

Let $\mathbb{K}$ be a number field of degree $n$, let $A = \mathcal{O}(\mathbb{K})$ be the ring of integers and consider the Minkowski embedding
$\mathbb{K} \rightarrow \mathbb{R}^n$ given by $x \...

**2**

votes

**0**answers

254 views

### Valuation topology vs modified valuation topology

Let $K$ be a field with valuation $v:K\to G\cup\{\infty\}$ where $G$ is an ordered abelian group. In section 7.62 of the book "Foundations of analysis over surreal number fields." Vol. 141. Elsevier, ...

**4**

votes

**1**answer

459 views

### p-adic expansion for elements in algebraic closure of p-adic numbers

In the following I will describe a proposal for the p-adic expansion of the elements of the algebraic closure $\overline{\mathbb{Q}_p}$ of $\mathbb{Q}_p$. My question is if this "conjecture" has been ...

**1**

vote

**0**answers

133 views

### Has there been proposed an extension of real numbers that connects logarithms and exponents in closed form? [closed]

Complex numbers do connect trigonometric functions with hyperbolic functions and exponents in closed form. Has anybody ever proposed an algebraic system that would connect in a similar way ...

**0**

votes

**1**answer

80 views

### Distinct projections along factors of splitting prime

Fix a number field $k$ and some $\alpha \in \mathcal{O}_k \setminus \mathbb{Z}$.
Let $p$ be a rational prime which splits completely in $k$, so that $p \mathcal{O}_k = P_1 \cdots P_m$ for $m = [k : \...

**11**

votes

**1**answer

442 views

### Upper bounds for regulators of real quadratic fields

We have an elementary sharp lower bound for the regulator of a real quadratic field as a function of the discriminant
$$R\geq \tfrac{1}{2}(\sqrt{d-4}+\sqrt{d})$$
It is sharp because the equality ...

**2**

votes

**0**answers

109 views

### relation between class number of an algebraic number field and its Galois closure

I read some results of C. Parry, F. Lemmermeyer and etc, about the class number formula in the special number fields, for example for any pure number field of a prime degree. Now I want to investigate ...

**1**

vote

**0**answers

66 views

### magma version of FixedGroup(K, L) over a different base field

I have a sequence of field extensions $F\subseteq L\subseteq K$ and I need to compute the Galois group of K over L. If $F=\mathbb{Q}$ then FixedGroup(K, L) does exactly this, but I was wondering if ...

**5**

votes

**0**answers

138 views

### free subgroups of $SL_2(\mathbb{Z[i]})$

The group $SL_2(\mathbb{Z})$ contains many free subgroups, for example all of the principal congruence subgroups for $n\geq 3$ and the subgroup $\left\langle \left(\begin{array}{cc}
1 & 2\\
0 &...

**8**

votes

**1**answer

409 views

### class number of prime degree field with prime conductor

Let $K$ be an finite abelian extension of $\mathbf{Q}$ conductor $p$, where $p$ is an odd prime. That is, $K \subset \mathbf{Q}(\mu_ p)$, the $p$-th cyclotomic field. Let $h_K$ be the class number of $...

**1**

vote

**0**answers

215 views

### Generalizing Dedekind's theorem on splitting of primes

Let $L/K$ be an extension of number fields. Suppose $\theta\in \mathcal{O}_L$ is a primitive element of this extension with $f(X)\in\mathcal{O}_K[X]$ its minimal polynomial over $K$.
Let $\mathfrak{p}...

**5**

votes

**0**answers

343 views

### 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 $p$ ...

**43**

votes

**2**answers

955 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 $\mathcal{...

**2**

votes

**1**answer

135 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 ...

**0**

votes

**0**answers

127 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?

**2**

votes

**0**answers

275 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 ...

**17**

votes

**2**answers

993 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 ...

**8**

votes

**1**answer

548 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

759 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} - ...

**33**

votes

**3**answers

1k 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 ...