Questions tagged [number-fields]
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74
questions with no upvoted or accepted answers
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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 ...
18
votes
1
answer
1k
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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 ...
12
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0
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418
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Is there a mistake in Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions"?
I am having troubles with Don Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions", available at this link, and I think there might be a mistake. In particular the ...
11
votes
0
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438
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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 ...
9
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0
answers
252
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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 ...
8
votes
0
answers
173
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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
$$\displaystyle S_K = \{\...
8
votes
0
answers
188
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The Stickelberger's annihilation theorem over an arbitrary number field
Let $p$ be a prime and let $C = \mathbb{F}_p^\times$. Then Gal$(\mathbb{Q}(\zeta_p)/\mathbb{Q})$, where $\zeta_p$ is a primitive $p$th root of unity, may be identified with $C$ in the obvious way. Let ...
6
votes
0
answers
496
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Genus of a number field
I'm reading Algebraic Number Theory by Neukirch. In chapter 3, he defines the genus of a number field as
$$ g = \log \frac{ |\mu (K)| \sqrt{|d_K|}}{2^{r} (2\pi)^{s}} $$ where $|\mu(K)|$ is its ...
6
votes
0
answers
681
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:...
5
votes
0
answers
297
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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 &...
5
votes
0
answers
706
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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$ ...
5
votes
0
answers
538
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 −...
4
votes
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123
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Factorization in the ring of integers of a particular biquadratic number field, and questions about norms
Consider the number field $K={\mathbb Q}[\sqrt{2},\sqrt{3}]$ and its ring of integers ${\mathcal O}_K$. I have been doing some calculations with this number field as a toy example, to see what can be ...
4
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0
answers
275
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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 ...
3
votes
0
answers
49
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On the complexity of global fields isomorphism
Let $L$ and $K$ be isomorphic global fields (i.e. either function fields of curves over a finite field, or number fields). What is the complexity of finding an isomorphism between them? Is there a ...
3
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0
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121
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Embedding number fields in fields with class number prime to $p$
Let $p$ be a fixed prime.
Question:
For any number field $K$, is there always a finite extension $L$ of $K$ of $p$-power order such that the class number of $L$ is prime to $p$?
Moreover, for any ...
3
votes
0
answers
126
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Congruence of elements implies congruence of norms for central simple algebras
I was reading Eichler's "Allgemeine Kongruenzklasseneinteilungen [...]", Crelle 1938 (one of the main historical references for strong approximation theorems), and I cannot understand one of ...
3
votes
0
answers
181
views
Decomposition of primes in cyclotomic extensions and their ramifications
Let $p$ be a prime. Suppose $L$ is a degree $p$ Galois extension over a number field $K$. Suppose $p$ splits both in $K$ and $L$.
So there will be $[K:\mathbb{Q}]$ primes of $K$ over $p$. Call them $...
3
votes
0
answers
109
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Furtwängler's family of irreducible polynomials
In the question Examples of nice families of irreducible polynomials over Z, user trew mentions a family of irreducible polynomials over the integers of the following form:
$$ p(x) = x^4 \prod_{i=1}^{...
3
votes
0
answers
505
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Fundamental Units in $\mathbb{Z}[\sqrt{d}]$ with $d \equiv 1 \mod 4$
It is well known and often cited how the fundamental units for the number ring of $\mathbb{Q}[\sqrt{d}]$ look like. In the case of $d\equiv 2,3 \mod 4$ the number ring is $\mathbb{Z}[\sqrt{d}]$ and in ...
3
votes
0
answers
126
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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 ...
3
votes
0
answers
104
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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 ...
3
votes
0
answers
302
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 $\sigma_{s+1},\overline{\sigma_{s+1}},\...
2
votes
0
answers
116
views
Imaginary quadratic fields with prime class number
Let $K$ be an imaginary quadratic field, with class number equal to an odd prime, say $h_K = p$.
In the proof Proposition 2.4 of this paper, Fukuda and Komatsu write,
"Since $h_K = p$, there ...
2
votes
0
answers
117
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$K(S,2)=\{b \in K^{\times}/{K^{\times}}^2\mid v(b)≡0 \bmod2, \forall v \notin S\}$ and Selmer group
This question is essentially related to the theory of elliptic curves (Selmer group), but this question itself is just a field theoretic one.
To calculate the Selmer group of given elliptic curve, we ...
2
votes
0
answers
103
views
Number fields with given discriminant
In the special case of number fields that are the splitting fields of irreducible polynomials of degree 5 or 6 with Symmetric Galois group (so degree 120 or 720 over Q), is there a good upper bound on ...
2
votes
0
answers
137
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Dirichlet unit theorem for finite rings
Let us fix a square free positive integer $n\in\mathbb{N}$ and consider the number field $\mathbb{Q}(\sqrt n)$ with ring of integers $K=\mathbb{Z}[\sqrt n]$. Let us denote the Galois norm of elements ...
2
votes
0
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99
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Weierstrass points in Artin-Schreier extensions of the rational function field in characteristic $p$
I encountered a problem when proving a result regarding the orders at non-Weierstrass points ($n$ is an order of the point $P$ if there is some holomorphic differential $\omega$ with order $n$ at $P$) ...
2
votes
0
answers
147
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Has there been much research on the Iwasawa theory of bi-quadratic fields?
The simplest non-trivial example of $\mathbb{Z}_p$-extensions happen over imaginary quadratic fields and I've seen a good amount of research done here (from my understanding still a lot to learn). I ...
2
votes
0
answers
122
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How do elliptic units generate the module of Euler systems over abelian extensions of imaginary quadratic fields?
I am trying to undesrtand the analogy between the Euler systems over abelian extensions of the rationals and the Euler systems over abelian extensions of imaginary quadratic fields.
As Soogil Seo ...
2
votes
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answers
110
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Iwasawa's results about relation between Galois cohomology and principal factorization
Let $K$ be a Galois number field with Galois group and units group $G$ and $U$, respectively. How we can relate the first cohomology group $H^1(G,U)$ to principal factorization in K?
I'd try to find ...
2
votes
0
answers
296
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Definition of Euler system of cyclotomic units
I am not sure about my understanding of Euler system of cyclotomic unit. This is what I have learnt:
Let $F=\mathbb{Q}(\mu_m)$.
Let $\mathcal{I}(m)$ = {positive square free integers divisible only by ...
2
votes
0
answers
91
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Elementary Iwasawa module
Let $k$ be a given number field. What is the importance and applications of knowing that the Iwasawa module $X_\infty$ of $k$ is an elementary $\Lambda$-module?
2
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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^...
2
votes
0
answers
204
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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 $\...
2
votes
0
answers
47
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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 ...
2
votes
0
answers
87
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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
550
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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, ...
2
votes
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160
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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 ...
2
votes
0
answers
342
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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 ...
2
votes
0
answers
489
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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 $Q(...
2
votes
0
answers
136
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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) = ...
2
votes
0
answers
246
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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 ...
2
votes
0
answers
194
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
0
answers
73
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Inflation-restrction sequence for maximal $S$-ramified extension
Let $K$ be a number field. Let $G_K$ be an absolute Galois group of $K$. Let $M$ be a $G_K$-module and $L/K$ be a finite extension.
There is a inflation-restriction exact sequence,
$0\to H^1(Gak(L/K), ...
1
vote
0
answers
148
views
The map from the decomposition group to the Galois group of the residue fields
$\DeclareMathOperator\Gal{Gal}$Let $L/K$ be a Galois field extension. Let $L$ be a field and $K$ be a number field. Let $B$ be a valuation subring of $L$ and let $A$ be the preimage of $B$ in $K$ (i.e ...
1
vote
0
answers
38
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Natural Density of Norms of ideals in a given ideal class
Some time ago, Landau proved the following formula for general number fields:
$I_K(x)=U_Kx+O(x^{\delta})$, where $\delta=1-2/(1+[K:\mathbb{Q}])$, where $I_K(x)$ is the number of ideals with norm below ...
1
vote
0
answers
74
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Liouville numbers with some "special" convergents
Recall that a real number $\ell$ is called a Liouville number, if there exists an infinite sequence of rational numbers $(p_n/q_n)_n$ for which
$$
0<\left|\ell-\frac{p_n}{q_n}\right|<\dfrac{1}{...
1
vote
0
answers
80
views
The localization map for the Mordell-Weil group of elliptic curves over finite Galois extensions
Let $L/K$ be a finite Galois extensions of number fields and $E/K$ be an elliptic curve. Denote by $\mathcal{F}$ the localization map
\begin{equation}
\mathcal{F}: H^1(G,E(L)) \rightarrow \bigoplus_{v ...
1
vote
0
answers
83
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How to describe the automorphisms of $\mathbb{Q}(\theta)$ that fix $\mathbb{Q}(\sqrt{-m})$
Suppose $m > 0$ is a square free integer and $m \equiv 1 $mod $4$. Then the $K = \mathbb{Q}(\sqrt{-m})$ is a subfield of $L = \mathbb{Q}(\theta)$, where $\theta$ is a primitive $4m$-th root of ...