Questions related to the algebraic structure of algebraic integers

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explicit example of computing ray class field for imaginary quadratic?

Given an imaginary quadratic number field K, we can get its ray class field mod some ideal $\mathcal{m}$ by adjoining the j-invariant of an elliptic curve with complex multiplication given by ...
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Help with proof regarding degrees of polynomials

How do you prove that if $f(x)\mid g(x)$ in $F[x]$, then either $g(x) = 0$ or $\deg(g(x)) \geq \deg(f(x))$ I'm not really sure how to prove these types of statements
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1answer
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Finding ideal representatives in the class group of $\mathbb{Q}(\zeta_{23})$

I know that $\mathbb{Q}(\zeta_{23})$ has class number 3, and I am wondering how I can find ideal representatives of the two nonprincipal classes in the class group. I have tried looking at examples ...
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1answer
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When Is there no Local Power Integral Basis?

Let $A$ be a Dedekind domain, $K, L, B$ the usual designations of $A$'s quotient field, a finite separable extension of $K$, and integral closure of $A$ in $L$ respectively. If $\alpha \in B$ ...
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2answers
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Finding non-negative integers $m$ such that $(1 + \sqrt{-2})^m$ has real part $\pm 1$.

I believe that the integers $m$ with $(1+\sqrt{-2})^m$ having real part $\pm 1$ are $0, 1, 2$ and $5$, but I'm having trouble proving it. Write $$a_m = \Re((1+\sqrt{-2})^m) = \frac{(1 + \sqrt{-2})^m ...
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1answer
55 views

Ring such that $q^2\mid p^2$ does not imply $q\mid p$?

Let $R$ be a commutative ring with $1$ and suppose $q^2\mid p^2,$ for $p,q \in R$. Unless $R$ is a UFD, I don't believe I can conclude that $q|p,$ but I would like to know a concrete counterexample. ...
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Fibonacci number that ends with 2014 zeros?

This problem is giving me the hardest time: Prove or disprove that there is a Fibonacci number that ends with 2014 zeros. I tried mathematical induction (for stronger statement that claims that ...
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1answer
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P-adic coefficient series [on hold]

Let $f$ be: $$f(x)=\sum_{n=0}^{\infty} a_nx^n$$ Such that $a_n \in \mathbb{Z_p}$ and $a_n \to 0$ We denote $|f|=\max|a_n| \space $and $\deg(f)=\max\{n:|a_n|=|f|\}$.$$$$ ($a$) Prove that if ...
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1answer
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Question on number theory ( related to (Z/p^rZ)* group )

This is a (different version to) question from Serre 'A Course in Arithmetic'.Let p be an odd prime number. $\forall n\geq 1$ (n positive integer), $f$ is defined by: $$f(n)=(-1)^n\prod_{1\le k\le n ...
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Can a ring of integers be free over a non-PID?

Let $K \subseteq L$ be an extension of number fields, and $A \subseteq B$ the corresponding rings of integers. $B$ is an $A$-module, generated by $[L : K]$ elements. If $K$ has class number one, ...
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Trivial Rost-Motive of a quadric

Let $q$ be the anisotropic,quadratic form of rank two corresponding to $\alpha = d(q) \in H^1(k,\mu_2)$. In his lecture notes "Topics in quadratic Forms" Vishik writes: For $n=1$ we get the ...
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1answer
21 views

Extensions of nonarchimedean valuations

This is a question from Janusz 'Algebraic Number Theory'. Let $R$ be a DVR with maximal ideal $\mathfrak p=\pi R$. Let $K$ be the quotient field of $R$ and $\mid\cdot\mid_{\mathfrak p}$ the ...
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1answer
38 views

Strong approximation theorem for Dedekind Domains

This is a theorem in "Maximal Orders" by Reiner. Page 48 stated without proof. And is said to be an easy consequence of The Chinese remainder Theorem. I am attempting to prove the theorem and need a ...
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2answers
63 views

Finding the GCD of two Gaußian integers

How do you calculate the GCD of $6-17i$ and $18+i$ in $\Bbb Z [i]$?
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2answers
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Rank($U$) = Rank($U^2$) for group of units $U$

I am reading the paper "Algebraic Integers on the Unit Circle" by Ryan C. Daileda (http://www.sciencedirect.com/science/article/pii/S0022314X05002027). I am confused about how he concludes that the ...
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Number of Primes in Ring of Integers of a Number Field

In the ring of integers of an algebraic number field, we say that an algebraic integer $p$ is prime if, whenever $p$ divides product of two algebraic integers, $p$ divides one of them. An algebraic ...
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Showing two numbers are relatively prime in number fields

Solve $x^3-2=y^2$ in integers. The standard way to solve this problem is to consider the arithmetic of the ring of algebraic integers $\mathcal{O}_{\mathbb{Q}(\sqrt{-2})}$ and to show that ...
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Do lattices in a field of fractions contain an ideal?

Let $R$ be a noetherian commutative integrally closed domain whose field of fractions $K$ is a finite extension of the field of fractions $Q$ of $\Lambda = \mathbb{Z}_p[[T]]$. Let $L \subset R$ be a ...
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1answer
45 views

A single-segment Newton polygon implies henselian?

I have a question about Newton polygons and henselian fields. In p149 of Neukirch’s book(algebraic number theory:the beginning of Proposition 6.7), he says that “We have just seen that the property ...
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1answer
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names in a quadratic field extension

Consider the quadratic field extension $\Bbb{Q}(\sqrt d)$. Is there a good name to use for the parameter $d$? Are there good names for $a$ and $b$ in the expression $a + b\sqrt d$? For example, ...
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1answer
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Uniquely determined discrete valuations

Let $K$ be a nonarchimedian discrete valued field. Let $f$ be a monic irreducible polynomial in $K[x]$. Let $w$ be an extended valuation to the splitting field of $f$. (The values of roots of $f$ are ...
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1answer
235 views

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$?

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$? I think there are no such polynomials, but how to prove?
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Characterizing a sequence of primes

This is an attempt to finish up Characterizing the primes which don't divide any Pell-Lucas number(s) For primes $p \equiv 3 \pmod 4,$ there is always some solution to $x^2 - 2 y^2 = \pm 1$ with ...
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1answer
80 views

Irrational to power of itself is natural

I've been thinking about a natural number like $n$ so that $x^x=n$ for some irrational $x$ but i couldn't find anything. As i didn't know how to approach the problem at all, i tried to make some ...
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1answer
70 views

How to prove this innocent looking isomorphism

I've got a Dedekind domain $R$ with quotient field $K$, a non-zero prime ideal $P$ of $R$. I form the completion $\widehat{K}$ of $K$ wrt the valuation $v_P$ associated to $P$. Let $\widehat{R}$ be ...
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3answers
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Is $\mathcal{O}_{\mathbb{Q}(\sqrt{5})} = \mathbb{Z}[\phi]$?

where $\phi$ is the golden ratio? I know that $5 \equiv 1 \mod 4$, so that then $\mathbb{Z}[\sqrt{5}]$ is not closed as far as integers go. But I'm a little confused cause ...
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1answer
99 views

Units of $\mathbb{Z}[\sqrt[4]2]$

How would one compute the units in $\mathbb{Z}[\sqrt[4]2]$? According to one source, it can be shown that the fundamental units are $1 + \sqrt[4]2$ and $1 + \sqrt{2}$, but it does not specify the ...
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2answers
119 views

Sets of Prime Numbers Generated By an Irreducible Monic Polynomial

Given a non-constant integral irreducible monic polynomial $f(x)$, the prime factors of its value at integers $x\in\mathbb{N}$ forms a set $\mathcal{P}(f)$. Is it possible that ...
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Modular Forms Over Quadratic Number Fields

I'm trying to do a bit of reading and looking into mislay forms of weight one over quadratic number fields, but am finding it difficult to locate any books or papers. I've got a little material from ...
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Algebraic closures are henselian?

Let $(K,v)$ be a nonarchimedean valued field and $(\widehat{K},\widehat{v})$ be its completion. Let $o$ and $\widehat{o}$ be the valuation rings of $K$ and $\widehat{K}$. Let $K_v$ be the separable ...
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3answers
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Is $\mathbb{Z}[\sqrt{2} + \sqrt{3}]$ integrally closed?

Or could it have a relation to another domain like $\mathbb{Z}[\sqrt{-3}]$ does to $\mathbb{Z}[\omega]$? Also, is it UFD? What are its units? I have never read about this domain in any book, though i ...
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Invariant of ramification under Mobius transformations

Is there some way of quantifying ramification of a cover up to Mobius transformations? i.e. If $K$ is an algebraically closed field of characteristic $p>0$, we can consider the Artin-Schreier ...
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1answer
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Examples where there is no power integral basis

Let $A$ be a completion discrete valuation ring with quotient field $K$, $L/K$ finite and separable, and $B$ the integral closure of $A$ in $L$. Let $P, \mathfrak P$ be the unique maximal ideals of ...
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Ring of integers is a PID but not a Euclidean domain

I have noticed that to prove fields like $\mathbb{Q}(i)$ and $\mathbb{Q}(e^{\frac{2\pi i}{3}})$ have class number one, we show they are Euclidean domains by tessalating the complex plane with the ...
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history and/or motivation for cohomology in class field theory

I am currently learning (local) class field theory via group cohomology with Milne's notes. I have a number of questions about using group cohomology to prove the main statements of class field ...
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A question in Neukirch's ANT book

In Corollary II.5.8, Neukirch Algebraic Number Theory(p142, line 11), why $d=v'_p(p)$ where $v'$ is normalized valuation? EDIT In other word, let $K$ be a finite extension of $Q_p$, I.e. a local ...
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1answer
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Haar Measure for Algebraic Number Theory: What Should I Know?

I recently taught myself some algebraic number theory and am preparing to take a course in class field theory this fall. I understand the notion of a Haar measure on a locally compact topological ...
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Meaning of tamely ramified extension.

Let $K$ be a complete field with respect to a discrete nonarchimedean valutaion. We denote $A$ and $\mathfrak{p}$ as its valuation ring and valuation ideal, respectively. For a finite Galois extension ...
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1answer
68 views

Why is the Ideal Norm Multiplicative?

I had asked this question before and got a partial answer. Let $A$ be a Dedekind domain with quotient field $K$, $L$ a finite separable extension of $K$ of degree $n$, and $B$ the integral closure of ...
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0answers
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Neukirch ANT's proofs for Hensel's Lemma and Extension Theorem for Valuations

I am in trouble with reading Neukirch, Algebraic-Number-Theory, Chap.II.4.6(Hensel's lemma). Question: Do we need the assumption of discreteness for nonarchimedian valuations in Hensel's Lemma(4.6) ...
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3answers
82 views

Real numbers that are not the roots of any polynomial equation with algebraic coefficients

An algebraic number is a number which is a root of some non-zero polynomial equation with rational coefficients. A transcendental number is a number which is not a root of any non-zero polynomial ...
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1answer
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Determine Units of a Ring $\mathbb{Z}[\alpha]$

I am trying to determine the units of $Z[\alpha]$ where $\alpha$ satisfies the monic polynomial $\alpha^4+\alpha^3+\alpha^2+\alpha+1$. I found $Z[\alpha] := \lbrace a+b\alpha+c\alpha^2+d\alpha^3\;|\; ...
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Rogers-Ramanujan Continued Fraction

How to calculate Rogers-Ramanujan Continued Fraction $R(e^{-2\pi{\sqrt{5}}})$ ?
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1answer
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Counting solutions mod p of a polynomial equation

Hello: Does somebody know if the following is true?: Let $f\in \mathbb{Z}[X]$ be a monic irreducible polynomial of degree $n$. Then there exists a positive integer $N$ and ...
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Rank these notations from most valid to least valid

$$\mathbb{Z}[\sqrt{-3}]$$ $$\mathbb{Z}\left[\frac{i \sqrt{3} - 1}{2}\right]$$ $${\bf{A}}(-3)$$ $$\mathbb{Z}[\omega]$$ $$\mathbb{Z}\left[-\frac{1}{2} + \frac{\sqrt{-3}}{2}\right]$$ ...
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Logarithm of the basic Lubin-Tate formal group

Let $K$ be a local field with finite residue field of cardinality $q$. Let $\pi$ be a uniformizer. The basic Lubin-Tate group (associated to $\pi$) is the unique formal group associated to the ...
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1answer
107 views

Changing streams in PhD

I've a masters degree from a reputed Indian university in pure mathematics, with a specialization in Algebraic Number Theory. However, I'd like to apply for a PhD in computational math/theoretical ...
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Problems in Algebraic Number Theory

if $z$ is an element of $Q(\zeta)$ ,where $\zeta$ is some $k$ th root of unity then $z^{(1/2)}$ is an element of $Q(\zeta^{(1/2)})$ ?
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Localization of lattices over Dedekind domains

I'm trying to understand the proof of lemma 4.12 on modules over Dedekind domains from Frohlich and Taylor's book 'Algebraic Number Theory' page 94. I have a Dedekind domain $\mathcal o$, a non-zero ...
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1answer
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How to show that сubed algebraic number is algebraic?

I have difficulty with that. But I can show that squared algebraic number is algebraic. Let $a$ be algebraic number. Than there is a polynomial $P(x)$ of power $n$ with integer coefficients (the ...