# Questions tagged [number-fields]

The number-fields tag has no usage guidance.

165
questions

**1**

vote

**0**answers

144 views

### Computing the kernel of some Artin-Map

let $K = \mathbb{Q}(\sqrt{-5})$ and $L = K(i)$. $\mathcal{O}_K$ is the ring of integers of K.
I would like to show that the kernel of the Artin-Map $\phi_{L/K}: I_K \rightarrow Gal(L/K)$ is $P_K$, ...

**-1**

votes

**1**answer

49 views

### Relative degree of a prime over a number field (notation from Algebraic Number Fields from Gerald J. Janusz) [closed]

I´m working with "Algebraic Number Fields" from Gerald J. Janusz (1. edition from 1973) and I have a question about his notation.
In chapter IV proposition 4.5 he states if K is an algebraic number ...

**2**

votes

**0**answers

59 views

### Cubic extensions of number fields and their local nature

Let $F$ be an irreducible squarefree cubic polynomial over a number field $K$. Let $L:=K[x]/{(F(x))}$ be a cubic extension of $K$.
Suppose that $\alpha \in L^\times$ such that $N_{L/K}(\alpha)=\beta^...

**3**

votes

**0**answers

84 views

### A question on p-rationality of number fields

Let $p$ be an odd regular prime and $F$ be a $p$-rational number field containing $\mu_p$. Equivalently, there is a unique prime $\mathfrak{p}$ above $p$ in $F$ and the $p$-class group is generated by ...

**3**

votes

**1**answer

276 views

### Integrality of Atkin-Lehner operator for $\Gamma_1(N)$

A result due to B. Conrad (http://math.stanford.edu/~conrad/papers/prasanna-inv.pdf, Theorem A.1) states that the Atkin-Lehner operator $w_{Q,k}$ is $\mathbb{Z}[1/Q]$-integral on $M_k(\Gamma_0(N))$. ...

**1**

vote

**1**answer

414 views

### Sets of primes with a given Frobenius conjugacy class

Let $a$ and $b$ be two relatively prime positive integers. Denote by $S$ the set of all positive primes of the form $a+bn$ (where $n$ is an integer, possibly zero or negative). Let $S'$ be any set of ...

**3**

votes

**0**answers

75 views

### Commutator subgroup of the absolute Galois group - a closed subgroup

Let $K$ be a finite extension of $\mathbb{Q}$. Is it possible that the commutator subgroup of the absolute Galois group of $K$ (considered as an abstract group) is a closed subgroup? This property ...

**3**

votes

**2**answers

273 views

### The kernel of the global class field theory homomorphism

Let $K$ be a finite extension of $\mathbb{Q}$. Then there is a surjective homomorphism $\theta:C_K\to G_K^{ab}$ from the idele class group to the abelianization of the absolute Galois group of $K$ (...

**1**

vote

**1**answer

282 views

### A type of principal ideal theorem of class field theory for ramified primes

Let $K$ be a number field and $\mathcal{O}_K$ be its ring of integers. Also let $p$ be a prime number, $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_K$ and $\zeta_{m}$ be a primitive mth root of ...

**1**

vote

**0**answers

87 views

### Octonion algebras over number fields [closed]

Is there any textbook or paper about Arithmetic of Octonion Algebras or Octonion Algebras constructed over number fields? I know J. Voight book and K. Martin notes about quaternion algebras but I was ...

**5**

votes

**0**answers

104 views

### $p$-torsion of class groups

Let $p$ be a fixed odd prime and $\ell$ be another prime such that $\ell \equiv 1 \pmod{p}$.
Consider the number field $\mathbb{Q}(\zeta_p)$ and its extension $\mathbb{Q}(\zeta_p, \zeta_\ell)$. Note ...

**1**

vote

**0**answers

133 views

### Matrix factorizations over $GL_2$ of a real quadratic ring of integers

tl;dr: The groups $GL_2(K)$, or $SL_2(K)$, where $K = \mathbb{C,R}$ admits several factorizations (the polar decomposition,
the KAN decomposition, the Schur triangular form, etc). Those
...

**1**

vote

**0**answers

66 views

### pari/gp “bnfisintnorm” as poor man (quadratic) Thue equations solver?

For simplicity explaining only the quadratic case.
Given integers $n,m$, pari/gp "bnfisintnorm" finds $X,Y$
such that $X^2+n Y^2=m$ working in the number field
with defining polynomial $x^2+n$ and ...

**1**

vote

**1**answer

121 views

### How to find a cyclotomic polynomial of degree d that decompose into d irreducible polynomials in $Z_6$?

More specifically, I need the degree $d$ to be around 1024. I can easily find the cyclotomic polynomial of degree 1024 that satisfies the above requirement in $Z_2$, i.e., $x^{1024}+1$, which is equal ...

**1**

vote

**0**answers

115 views

### Terminology about ramification

Let $K$ be a totally real (finite) number field. Let $S$ be a finite set of places of $K$ containing the primes above a prime number $p$. Let $K_S$ be the maximal abelian extension unramified outside $...

**2**

votes

**3**answers

210 views

### Mahler measures of values of polynomials

Let $K\ne \mathbb{Q}$ be a number field, let $\alpha\in \mathcal{O}_K$ and let $f(X)\in \mathcal{O}_K[X]$. Denote the Mahler measure by $M$.
Is there any known result about the comparison of the ...

**3**

votes

**0**answers

139 views

### Dihedral extension of $\mathbb Q$ with small discriminant

Let $K$ be a fixed quadratic number field, say $K=\mathbb Q(\sqrt 5)$. For any integer $n \geq 3,$ I would like to build a number field $D_n$ such that $D_n/\mathbb Q$ is Galois, with Galois group ...

**11**

votes

**1**answer

1k views

### Is there an elementary proof that there are infinitely many primes that are *not* completely split in an abelian extension?

I'm currently in the middle of teaching the adelic algebraic proofs of global class field theory. One of the intermediate lemmas that one shows is the following:
Lemma: if L/K is an abelian ...

**3**

votes

**1**answer

218 views

### Fermat's cubic equation in quadratic extension of $\mathbb{Q}$

Is still relevant or interesting be capable to bring a criteria in order to classifly quadratic extensions of $\mathbb{Q}$ based on the existence or not existence of non-trivial solutions of Fermat's ...

**9**

votes

**0**answers

187 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

108 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

403 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 ...

**3**

votes

**0**answers

92 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

72 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

643 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/...

**7**

votes

**1**answer

347 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 ...

**13**

votes

**1**answer

290 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

283 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:...

**3**

votes

**0**answers

210 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

665 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

113 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

131 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

854 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

305 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

333 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

98 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

182 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

65 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

397 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

162 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

75 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

756 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

344 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

167 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

188 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

149 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

206 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

177 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

61 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 \...