# Questions tagged [number-fields]

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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 $\mathfrak{A}_{L/\mathbb{Q}}=\{x\in \... 0answers 68 views ### Norm groups of number fields I came across this proposition in an article about genus class fields. I have a few questions about the parts that I have underlined in red. I don't understand why the norm map$N_{H/K}: I_H \to P_K$... 0answers 74 views ### Weierstrass points in Artin-Schreier extensions of the rational function field in characteristic$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$) ... 0answers 101 views ### Has there been much research on the Iwasawa theory of bi-quadratic fields? The simplest non-trivial example of$\mathbb{Z}_p$-extensions happen over imaginary quadratic fields and I've seen a good amount of research done here (from my understanding still a lot to learn). I ... 1answer 443 views ### How Dirichlet proved Dirichlet's unit theorem for general number fields? For a general number field$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$. It seems that standard algebraic ... 0answers 94 views ### What conditions on primes makes it non principal? Let$K$be a number field and$\mathfrak{p}$is prime of$K$what condition I can have on$\mathfrak{p}$so it becomes non-principal ideal? 0answers 79 views ### How to determine if a unramifed prime split or not? Let$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}$splits completely in ... 0answers 71 views ### A subgroup of the$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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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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### 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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### 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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### 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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### $G_{\mathbb Q}$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### $p$-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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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 \$\...