Questions related to the algebraic structure of algebraic integers

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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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I propose the following difinition, a number of “hyper-transcendent” [on hold]

We say that a number x is hyper-transcendent, if it can not be the root of any equation: f (x) = 0, since the function F, and a combination of a finite number of elementary functions, and the ...
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1answer
41 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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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
79 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
64 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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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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1answer
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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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23 views

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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27 views

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
30 views

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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2answers
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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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1answer
29 views

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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1answer
62 views

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
67 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
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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
66 views

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
50 views

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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2answers
134 views

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
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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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1answer
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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 ...
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1answer
187 views

Is $\mathbb{Z}[\sqrt{2} + \sqrt{3}]$ closed under multiplication?

Bonus question: if it's not, is it a subdomain of some ring of algebraic integers? This is just something I was thinking about a few weeks ago. I forgot about the concept of algebraic degrees, which ...
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5answers
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How to tell if a Fibonacci number has an even or odd index

Given only $F_n$, that is the $n$th term of the Fibonacci sequence, how can you tell if $n \equiv 1 \mod 2$ or $n \equiv 0 \mod 2$? I know you can use the Pisano period, however if $n \equiv 1$ or ...
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4answers
126 views

When is the sum of two squares the sum of two cubes

When does $a^2+b^2 = c^3 +d^3$ for all integer values $(a, b, c, d) \ge 0$. I believe this only happens when: $a^2 = c^3 = e^6$ and $b^2 = d^3 = f^6$. With the following exception: $1^3+2^3 = 3^2 + ...
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Wolstenholme Number

Does Wolstenholme Numbers have perfect squares other than 1 and 49? The first few are 1, 5, 49, 205, 5269, 5369, 266681, 1077749 seems to be a complicated problem
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Neukirch Algebraic NT 12.12

I do not understand Theorem (12.12) at page 81 in Neukirch, Algebraic Number Theory. I can verify each line of the proof, but the following is so clear? {\bf Problem} Let $o$ be an order in an ...
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1answer
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Valuation associated to a non-zero prime ideal of the ring of integers

I have a question from Frohlich & Taylor's book 'Algebraic Number Theory', p.64. I will keep the notation used there. Let $K$ be a number field, $\mathcal o$ its ring of integers. Let $\mathfrak ...
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1answer
30 views

Expressing $\mathcal O_L$ as a certain free module of rank 1

I have a finite Galois extension of number fields $L/K$ with group $G$ and respective rings of integers $\mathcal O_L$ and $\mathcal O_K$. If $\Gamma$ is an $\mathcal O_K$-order in $K[G]$ and ...
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A finite Galois extension $L/K$ of number fields.

I have a finite Galois extension $L/K$ of number fields with group $G$. Let the respective rings of integers be $\mathcal O_L$ and $\mathcal O_K.$ Suppose that $\Gamma$ is an $\mathcal O_K$-order in ...
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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 ...
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The structure of $O_L$ as an $O_K$ module.

This a a second thought after the question: Is $O_L$ a free $O_K$ module? So, if $L/K$ is a finite number field extension,I know that we can find $\beta_1,\dotsc,\beta_n\in L$ such that ...
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1answer
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Is $O_L$ a free $O_K$ module? [duplicate]

This must have been known. Let $L/K$ be a finite number field extension. Then is $O_L$ a free $O_K$ module? How to prove it if so? So far, I know how to prove the following: Let ...
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0answers
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A certain $\mathcal O_K$-lattice where $K$ is a number field

I have a finite Galois extension of number fields $L/K$ with group $G$. Let $\mathcal O_L$ and $\mathcal O_K$ be the respective rings of algebraic integers. I want to show that $\mathcal O_L$ is ...
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1answer
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Rings of algebraic integers

A basic question on algebraic numbers. If $L/K$ is a finite extension of number fields with respective rings of integers $\mathcal O_L$ and $\mathcal O_K$ then is it true that $\mathcal O_L$ is ...
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1answer
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Serre's Modularity Conjecture — Weight

I was reading Serre's paper "Sur les Représentations Modulaires de Degré $2$ de Gal($\bar{\mathbb{Q}}/\mathbb{Q}$)" where he states his modularity conjecture (which is now a theorem). Following his ...
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Why is the norm of an ideal contained in that ideal?

Suppose $K$ is a number field and that $\mathcal{O}_K$ is the ring of integers of $K$. Now, let $I$ be an ideal in $\mathcal{O}_K$. I know that $N(I) \in I$, but I want to prove it. By definition, ...
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3answers
65 views

Prove some number is algebraic over a field

How do you prove (without calculating the minimum polynomial) that $\sqrt{3}$ + $\sqrt[]{5}$ is algebraic over $\mathbb{Q}$. Also prove that $\left(\mathbb{Q}(\sqrt{3} + \sqrt[]{5}\right):\mathbb{Q}) ...