Questions tagged [number-fields]

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Dihedral extension unramified at primes dividing order of group?

Consider dihedral Galois extensions $L/\mathbb{Q}$ of degree $n$ (and we know they exist thanks to Shafarevich), can we show there always exists an extension $L/\mathbb{Q}$ unramified at $p$, for all $...
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1 vote
0 answers
56 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 ...
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1 vote
0 answers
76 views

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 ...
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2 votes
0 answers
65 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 ...
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3 votes
0 answers
116 views

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I was reading Eichler's "Allgemeine Kongruenzklasseneinteilungen [...]", Crelle 1938 (one of the main historical references for strong approximation theorems), and I cannot understand one of ...
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  • 697
4 votes
1 answer
156 views

Order of $37$-Sylow subgroup of ideal class group of $K_{37} = \Bbb Q(\mu_{37^{n}})$ is known to be $37^n$

I want to examine nontrivial examples of what we call Iwasawa class formula, $c(n)=\mu p^n + \lambda n + \nu$, where $\lambda, \mu \in \mathbf N$ and $\nu \in \mathbf Z$ are parameters depending only ...
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  • 141
4 votes
1 answer
242 views

Common Galois extension over $\mathbb Q $

Suppose $L'$ is a fixed cyclic galois extension over $\mathbb {Q} $ of degree $4$.Now we know that there exists also a degree $k$ extension $L$ over $L'$ but the extension $(L/\mathbb{Q})$ may not ...
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  • 693
3 votes
0 answers
96 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 $...
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4 votes
0 answers
91 views

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 ...
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2 votes
0 answers
119 views

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 ...
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3 votes
0 answers
89 views

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}^{...
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  • 125
1 vote
0 answers
87 views

Translation of a paper by Dedekind (an integral basis for pure cubic fields)

I am studying Introductory Algebraic Number Theory written by S. Alaca and K. Williams. The authors mention the theorem concerning an integral basis for pure cubic fields but do not provide proof. ...
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9 votes
1 answer
544 views

Is it true that this ideal must be principal? (proof verification)

Let $L/K$ be a (abelian, Galois) quadratic extension of number fields with $\text{Gal}(L/K)$ generated by $\sigma$ and $\mathfrak{p} = \alpha\mathcal{O}_K$ a principal prime ideal of $\mathcal{O}_K$. ...
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  • 287
2 votes
1 answer
132 views

Does a quaternion algebra exist over a number field that is split over some infinite real places, but not others?

Let $F$ be a totally real number field having at least two different real embeddings $\sigma_1 : F \hookrightarrow \mathbb{R}$ and $\sigma_2 : F \hookrightarrow \mathbb{R}$. Does a quaternion algebra $...
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3 votes
0 answers
275 views

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 ...
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6 votes
1 answer
284 views

Galois module theory: from global to local

Let $L/\mathbb{Q}$ be a finite Galois extension with Galois group $G$. It is well known that the ring of integers $\mathcal{O}_L$ is free over its associated order $\mathfrak{A}_{L/\mathbb{Q}}=\{x\in \...
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4 votes
1 answer
126 views

Norm groups of number fields

I came across this proposition in an article about genus class fields. I have a few questions about the parts that I have underlined in red. I don't understand why the norm map $N_{H/K}: I_H \to P_K$ ...
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  • 529
2 votes
0 answers
78 views

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 $P$) ...
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2 votes
0 answers
127 views

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 ...
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13 votes
1 answer
775 views

How Dirichlet proved Dirichlet's unit theorem for general number fields?

For a general number field $K$, Dirichlet's unit theorem states that the unit group of the ring of integers of $K$ is a finitely generated group of rank $r_1+r_2-1$. It seems that standard algebraic ...
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  • 741
1 vote
0 answers
100 views

What conditions on primes makes it non principal?

Let $K$ be a number field and $\mathfrak{p}$ is prime of $K$ what condition I can have on $\mathfrak{p}$ so it becomes non-principal ideal?
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  • 333
0 votes
0 answers
82 views

How to determine if a unramifed prime split or not?

Let $K$ be the Number field and $L$ be finite extension where $\mathfrak{p}$ prime of K is unramified. Are there any conditions on $\mathfrak{p}$ so that I can say $\mathfrak{p}$ splits completely in ...
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1 vote
0 answers
82 views

A subgroup of the $n$-Selmer group

Let $p$ be an odd prime and for the purpose of this question let $n$ be an integer which is a power of $p$. Let $E$ be an elliptic curve over a number field $F$. The $n$-Selmer group, denoted by $S_n(...
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  • 1,059
4 votes
2 answers
205 views

Richaud-Degert type quadratic extensions

A Richaud-Degert type real quadratic field is a number field of the form $K = \mathbb{Q}(\sqrt{d})$ where $d = {(an)}^2 + ka > 0$ for positive integers $a, n$ and $k \in \{ \pm 1, \pm 2, \pm 4 \}$, ...
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  • 529
1 vote
0 answers
59 views

Exponents of powers in cyclotomic extensions

For every number field $K$ and for every $a \in K$ let $$\epsilon(K;a) := \sup\!\left\{e \in \mathbb{N} : \exists b \in K \text{ such that } a = b^e\right\}. $$ Now let $K$ be a number field and let $...
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  • 11
8 votes
1 answer
224 views

What is the lowest complexity definition of $\mathbb{Z}$ in an infinite extension of $\mathbb{Q}$

In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. And in 2012, Jennifer Park proved a result which implies that $\mathbb{Z}$ is $\exists\forall$-...
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5 votes
1 answer
141 views

On the stabilizer in $\mathrm{GL}(2,\mathbb{Z})$ of a real quadratic irrationality

$\DeclareMathOperator\GL{GL}$Let $\theta$ be a real quadratic irrationality with discriminant $\Delta$; let $\mathcal{O}_\Delta$ denote the resulting quadratic order of discriminant $\Delta$ in $\...
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3 votes
2 answers
316 views

Correspondence between binary quadratic representations and proper ideals of quadratic number fields

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}$Fix $d < 0$, a fundamental quadratic discriminant and $n$ a positive integer. Suppose $Q$ is a primitive binary quadratic form of ...
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  • 529
2 votes
0 answers
99 views

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 ...
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  • 99
1 vote
0 answers
330 views

Looking for a paper of Lagarias and Odlyzko

I have been studying about the Chebotarev Density Theorem and have been hunting for the following paper of Lagarias and Odlyzko for quite a while: Effective versions of the Chebotarev density theorem, ...
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9 votes
1 answer
321 views

Standard conjecture on u-invariants?

This is well beyond my expertise, but I just learned some of the history behind $u$-invariants of fields $F$, where ($u(F)+1$)-variable quadratic equations always have a non-trivial solution, but $u(F)...
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0 votes
1 answer
155 views

English translation of Hasse's "Arithmetische Theorie der kubischen Zahlkorper auf klassenkorpertheoretischer Grundlage"

I want to read through Hasse's paper about cubic number fields: Arithmetische Theorie der kubischen Zahlkorper auf klassenkorpertheoretischer Grundlage, Mathematische Zeitschrift 31 (1930) pages 565-...
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  • 529
2 votes
0 answers
88 views

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 ...
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2 votes
0 answers
162 views

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 ...
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  • 99
0 votes
1 answer
189 views

Logarithmic Weil height

Let $a_0,\cdots,a_n$ be algebraic integers. Is $h(a_0,\cdots,a_n)\le\max_{0\le i\le n}\log(\max(1,|a_i|))$ where $h(a_0,\cdots,a_n)$ denotes the logarithmic Weil height? Thanks in advance.
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  • 2,997
1 vote
0 answers
163 views

Do roots of polynomial with coefficients in a CM field lie in a CM field?

This is something that I have been thinking about for a while now, not sure if it is standard (or even true at all) or not: Let $K/ \mathbb Q$ be a CM number field, that is, it is closed under complex ...
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9 votes
1 answer
280 views

Galois embedding question for dihedral groups

Let $D_n$ be the dihedral group of order $2n$. Then all the quotients of $D_n$ are dihedral as well, and of the form $D_k$ with $k \mid n$. So for a field $K/\mathbb{Q}$ with $\operatorname{Gal}(K/\...
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  • 91
1 vote
1 answer
232 views

$G_{\mathbb Q}$ and primes of $\overline{\mathbb{Z}}$

I know that if $K/\mathbb Q$ is a finite Galois extension (i.e. a Galois number field), then for any prime $(p)\subseteq \mathbb Z$, the Galois group $G=\operatorname{Gal}(K/\mathbb Q)$ acts ...
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0 votes
0 answers
46 views

prime decomposition in even dihedral extensions

Let $L/K$ be a finite extension of number fields of degree $n$ with $n$ an even integer such that the normal closure of $L$ has the Galois group isomorphic to $D_n$, the dihedral group of order $2n$. ...
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3 votes
1 answer
169 views

Existence of algebraic integer with absolute value equal to reciprocal of maximum of $1$ and absolute value of a given algebraic number

Consider a number field $K$, and let $v_1, \cdots v_n$ ($n \in \mathbb N$) be some finite (i.e. non-archimedean) places of $K$. Is the following true? For every $\alpha \in K^\times$ there exists $\...
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2 votes
0 answers
78 views

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?
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  • 21
1 vote
0 answers
198 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 $P_K$, ...
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  • 13
-1 votes
1 answer
69 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 ...
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  • 13
2 votes
0 answers
80 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^...
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4 votes
1 answer
329 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 ...
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  • 1,059
3 votes
1 answer
316 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 $M_k(\Gamma_0(N))$. ...
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1 vote
1 answer
452 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 ...
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8 votes
1 answer
316 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 ...
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3 votes
2 answers
404 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 $K$ (...
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1 vote
1 answer
429 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 ...
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