Questions related to the algebraic structure of algebraic integers

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631 views

Serge Lang Never Explains Anything Round II

I'm reading the second edition of Lang, Algebraic Number Theory, page 221. I quote: Let $F$ be a local field, i.e. the completion of a number field at an absolute value. Let $L$ be an abelian ...
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422 views

Which number fields can appear as subfields of a finite-dimensional division algebra over Q with center Q?

I have some idle questions about what's known about finite-dimensional division algebras over $\mathbb{Q}$ (thought of as "noncommutative number fields"). To keep the discussion focused, let's ...
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68 views

Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$.

We consider an analogue of the Dirichlet $L$-function in $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, ...
12
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177 views

CFT via Brauer groups vs via ideles

I am interested in the relationship between the following two versions of CFT: Version 1: (Brauer Group Version) Let $K$ be a number field. One constructs, for every finite place $v$ of $K$, a map ...
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111 views

Proving that the number of integer solutions of $x^2-Ny^2=1$ is infinite

I am trying to prove that the number of integer solutions of $x^2-Ny^2=1$ is infinite whenever N is a squarefree integer. For this I define norm of $a+b\sqrt N=a^2-Nb^2$. Now I prove that $a+b \sqrt ...
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506 views

extension of Euler's totient function to number fields

It is well known that the Euler totient function $\varphi$ satisfies the formula $n = \sum_{d | n}\varphi(d)$. This follows for example from the fact that $\mathbb Z / n \mathbb Z$ can be written (as ...
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89 views

Classification of all subrings

Let $R$ be an integral domain whose underlying additive group is finitely generated free and whose field of fractions $K$ is a finite Galois extension of $\mathbb{Q}$. Is there a method of ...
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647 views

A proof of a theorem on the different in algebraic number fields

Theorem Let $K ⊂ L ⊂ E$ be a tower of algebraic number fields. Suppose that $E/K$ is a Galois extension. Let $B$ and $C$ be the rings of algebraic integers in $L$ and $E$ respectively. Let $G$ be the ...
8
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82 views

Genus of $k(T)$?

Let $F$ be a function field in one variable with total constant field $k$, let $X$ be the set of all places $F$, and let $S$ be a nonempty finite subset of $X$. Then the genus of $F$ is equal to the ...
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206 views

Irreducibility of cyclotomic polynomials via schemes

A few months ago, someone told me there existed a scheme theoretic proof of the irreducibility of cyclotomic polynomials. I've tried coming up with a proof, and when that didn't really yield anything ...
8
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133 views

Specializations of elementary symmetric polynomials

Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$ indeterminates. The $h^{th}$elementary symmetric polynomial is the sum of all monomials with $h$ factors \begin{eqnarray*} ...
8
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238 views

Numbers represented by a cubic form

EDIT, April 11, 2013: See answer at http://mathoverflow.net/questions/127160/numbers-integrally-represented-by-a-ternary-cubic-form/127295#127295 This is part 2 ( of 25 discriminants of class number ...
8
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210 views

Is the splitting field unramified over a prime $\mathfrak{p}$ if the discriminant of the polynomial is coprime to $\mathfrak{p}$?

Is the following statement true? Let $R$ be a discrete valuation ring with quotient field $K$ and valuation $\nu$. Suppose that $f(x)\in K[x]$ is an irreducible separable polynomial with ...
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103 views

What does the ideal norm of matrix elements really mean?

Say we have a number field $K$ (specifically, an imaginary quadratic field) and a $2\times2$ matrix $\sigma=\pmatrix{a&c\\b&d}$ with elements $a,b,c,d\in\mathcal O_k$, the ring of integers of ...
7
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125 views

Name of a certain set

I want to know if there is any already-standard way to refer to the sets described as follows. For a set $X$, let $-X = \{-x: x \in X \}$; call it the negative of $X$. Take the set of all primes in ...
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When exactly is the splitting of a prime given by the factorization of a polynomial?

Let $L/K$ be an extension of number fields with $L=K(\alpha)$, where $\alpha\in \mathcal{O}_L$. Let $f\in \mathcal{O}_K[x]$ be the minimal polynomial of $\alpha$ over $K$. Let ...
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Ex. $2.30$ in Silverman Adv. Topics

I would like to refer you to Exercise $2.30(c)$ in Silverman's Advanced Topics in Elliptic Curves. Question: Let $E/L$ be an EC with CM by $K$. Assume that $K\nsubseteq L$, and let $L'=LK$ and let ...
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506 views

A conjecture about Lucas series

Let $L_n$ be the Lucas series: $L_0=2,L_1=1,L_n=L_{n-1}+L_{n-2}(n>1).$ $p$ is a prime number and $p\equiv3,7\pmod {20}$, hence $\exists x,y\in \mathbb Z:2p=x^2+5y^2.$ Is it true that ...
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573 views

Compute the Unit and Class Number of a pure cubic field $\mathbb{Q}(\sqrt[3]{6})$

Find a unit in $\mathbb{Q}(\sqrt[3]{6})$ and show that this field has class number $h=1$. I am done with the first part which is relatively simple: Suppose that $\varepsilon$ is a unit in ...
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391 views

notation for ramification index and inertial degree

For a prime $Q$ lying over a prime $P$, I have seen the ramification index of $Q$ over $P$ denoted by $e(Q|P)$ and the inertial degree of $Q$ over $P$ by $f(Q|P)$. What is the origin of the ...
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160 views

closure of units of number fields in the finite idele topology

Let $K$ be a number field. Denote by $\mathcal O _K^\times$ its rings of units and by $\mathcal O _{K,+} ^\times$ its ring of totally positive units. Further let us denote by $\mathbb A ...
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36 views

Given $h(\mathcal{O}_{\mathbb{Q}(\sqrt{d})}) = 1$, what is the longest possible run of inert primes in that ring?

Say $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ is a unique factorization domain. A couple of primes will ramify, the rest will either split or be inert. For example, in ...
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99 views

Unique factorisation in ring generated by square roots of prime numbers?

Let $\langle\sqrt{\mathbb{N}}\rangle=\mathbb{Z}[\sqrt2,\sqrt3,\sqrt5,\ldots]$ denote the ring generated by the square roots of all prime numbers. Is it it known whether ...
6
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177 views

Generalized class group of $\mathbb Q(\sqrt{-5})$

I follow the notation of Georges Gras: Class Field Theory, some of which I recall for convenience; feel free to skip the following lines if you are familiar with the notation. Let $K$ be a number ...
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28 views

Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ , $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$, what is the intersection?

Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ and $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$, What is the intersection $F_\infty\cap K_\infty$? Here $\zeta_{2^n}$ is a ...
6
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120 views

27 lines on a smooth cubic surface

It is known that every smooth cubic surface with coefficients in $\mathbb{Q}$ has $27$ lines defined over a number field extension of $\mathbb{Q}$ of degree at most $51840$ as the group ...
6
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92 views

Ramification index of infinite primes

I am reading Neukirch's Algebraic Number Theory. On page 184, Chapter 3, Neukirch defines the ramification index of infinite primes as follows: For a finite extension $L/K$ of number fields, and an ...
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217 views

Generalization of a result on finite Galois extensions to infinite case.

It is well known that if $L/K$ is a finite abelian extension, $\frak p$ a prime of $K$ and $\frak P$ a prime of $L$ above $\frak p,$ then $L^I /K$ is the maximal subextension of $L/K$ in which ...
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193 views

Coefficients in expansion of $(\sqrt[3]{2} - 1)^m$

In trying to solve $a^3 - 2b^3 = 1$ over the integers I came across the need to answer the question: when does $(1+ \sqrt[3]{2} + \sqrt[3]{2}^2)^n$ have no $\sqrt[3]{2}^2$ term in it's expansion (in ...
6
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102 views

Classes of groups known to be realizable (IGP)

A finite group $G$ of order $n$ is said to be realizable (over $\mathbb{Q}$) if there exists a Galois extension $L/\mathbb{Q}$ such that $\mathrm{Gal}(L/\mathbb{Q})=G$. I'm curious what classes of ...
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77 views

Generalization of Kummer isomorphism?

Let $p$ be a prime number and denote by $\mathbb{F}_p(1)$ the one dimensional vector space over $\mathbb{F}_p$ endowed with an action of $G:=Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p)$ via the mod $p$ ...
6
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242 views

Splitting of a polynomial modulo primes of a ring of integers

I'm trying to finish an exercise from Daniel Marcus's "Number Fields" book: #30(e) in chapter 3, page 91. Here's the problem: say $\mathcal{O}_K$ is the ring of integers of a number field $K$ and let ...
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174 views

The generalized Euler's function in number field for ideal $\mathfrak{a}$

Is there any results about the Euler's totient function for ideals? More precisely, for any number field $k,$ if we define $\phi(\mathfrak{a})$ as the number of residue classes ...
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133 views

Hecke characters and Unitary groups

Let $M/F$ be a quadratic extension of number fields, with Galois group $G=\{1,\tau\}$. Consider the following unitary group $$U_1(R)=\{z\in (R\otimes_FM)^\times :zz^\tau=1\},$$ where $R$ is an ...
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31 views

Are the ring of integers of the constructible numbers a Euclidean domain?

I suspect that since Euclid uses the Euclidean Algorithm to perform division on constructible numbers in Elements, the ring of integers of the constructible numbers are a Euclidean Domain, but I have ...
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231 views

rationality of $\ell$-adic representation attached to an elliptic curves

Let $E$ be an elliptic curves defined over a number field $K$. Consider the $\ell$-adic representation attached to $E$ $$ \rho_{\ell}:\mathrm{Gal}(\overline{K}/K) \longrightarrow ...
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75 views

Neukirch's motivation for $p$-adic numbers

I've started reading Neukirch's Algebraic Number Theory book and at the beginning of Chapter II he starts his motivation for the $p$-adic numbers as follows: "The idea originated from the observation ...
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Connection between sgn character and the Legendre symbol

Today, while I was lecturing on the Legendre symbol, I realized that the phenomenon: "the product of two non-squares is a square" isn't so foreign. For example, for $\mathbb R^\times$, the squares are ...
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An infinite prime can ramify right? (So what is Neukirch talking about?)

I have been under the impression for several years that if $L/K$ is an extension of number fields, then an infinite place of $K$ is said to ramify in $L$ if it comes from a real embedding of $K$ which ...
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110 views

Prime number theorem for Dedekind domains

Let $\mathscr P\subseteq \mathbb N$ be the set of prime numbers. The prime number theorem tells us that if $\pi(x)=|\{p\in\mathscr P\colon p\leq x\}|$ then $\pi(x)\sim \frac{x}{\log x}$. Now one could ...
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Difficulty of exercises in several books

Since I am teaching myself, I have no idea of the difficulties of exercises in many books. I want to know whether I am not understanding the content of these books if I cannot easily solve the ...
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91 views

Isomorphism of algebras $E_v = \prod_{w\mid v} E_w$

I am currently reading the article Bayer–Fluckiger, Eva, B. Nivedita, and Raman Parimala. "Hasse principle for G–quadratic forms." Documenta Mathematica 18 (2013): 383-392. in which there is a ...
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164 views

Given an integer, how can I detect the nearest integer perfect power efficiently?

If you give me an integer N, how can I detect the nearest integer perfect power, larger or smaller than N? In other words, the perfect power the distance between N and which is less than the ...
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50 views

Is the group of principal ideles closed in the group of finite ideles?

Let $K$ be an algebraic number field. Let $\mathbb{I}$ be the group of ideles and let $\mathbb{I}_f$ be the group of finite ideles. We embed $K^\times$ diagonally in both. It is know that $K^\times$ ...
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Profinite group power map

I am working through some exercises in Neukirch's Algebraic number theory and i need some hints for exercise 1 pg 274, it goes as follows: Let $G$ be a profinite group, show that we can extend the ...
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140 views

Question about a proof on p70 in Cassels' Local Fields

I'm trying to read the proof of COROLLARY. The only solutions of $x^2+7=2^m$ ($x,m \in \mathbb{Z}$) (6.15) have $m=3,4,5,7,15$. I don't see why there could be a + in $y\pm \alpha$ ...
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380 views

Picard group of an order in an imaginary quadratic field as a quotient of the idele class group

If $K$ is an imaginary quadratic field, then one can show that the orders in $K$ are precisely the rings $\mathbb{Z}+f\mathscr{O}_K$ for $f\geq 1$, where $\mathscr{O}_K$ is the ring of integers of ...
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110 views

Ramification of an integral closure of $\mathbb{C}\{z\}$

Let $\mathbb{C}\{z\}$ be the ring of convergent series in one variable over $\mathbb{C}$, $K$ the fraction field of $\mathbb{C}\{z\}$, $E$ a Galois extension of $K$ and $\mathcal{O}_{E}$ the integral ...
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172 views

Is there a notion of *p-adic Dedekind Domains*?

As we all know, the ring $Z_p$ can be constructed as the projective limit of the rings $Z/p^{n}Z$. Now is there any generalization such as the p-adic completions of a Dedekind Domain? This might be ...
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160 views

Is there an integral domain with a lot of residue fields of the same characteristic?

Is there a commutative integral domain $R$ in which: every nonzero prime ideal $Q$ is maximal, and for every prime power $q\equiv 3 \bmod 8$, there is a maximal ideal $Q$ of $R$ such ...