Questions related to the algebraic structure of algebraic integers

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3
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2answers
82 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
29 views

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 ...
8
votes
2answers
62 views

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 ...
3
votes
1answer
47 views

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 ...
3
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0answers
19 views

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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vote
2answers
78 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 ...
3
votes
1answer
48 views

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, ...
1
vote
1answer
18 views

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 ...
23
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1answer
273 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?
7
votes
1answer
131 views

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 ...
5
votes
1answer
88 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 ...
3
votes
1answer
76 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 ...
6
votes
3answers
85 views

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 ...
5
votes
1answer
106 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 ...
10
votes
2answers
127 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 ...
2
votes
0answers
40 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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votes
0answers
31 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 ...
5
votes
3answers
136 views

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

Is $\mathbb{Z}[\sqrt{2} + \sqrt{3}]$ integrally closed? (Or could it have a relation to another domain like $\mathbb{Z}[\sqrt{-3}]$ does with $\mathbb{Z}[\omega]$?) Also, is it UFD? What are its ...
1
vote
0answers
28 views

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 ...
3
votes
1answer
44 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 ...
11
votes
2answers
373 views

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 ...
5
votes
2answers
83 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 ...
0
votes
1answer
71 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 ...
6
votes
1answer
86 views

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 ...
2
votes
0answers
33 views

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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vote
1answer
86 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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vote
0answers
24 views

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) ...
4
votes
3answers
109 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 ...
3
votes
1answer
89 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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0answers
56 views

Rogers-Ramanujan Continued Fraction

How to calculate Rogers-Ramanujan Continued Fraction $R(e^{-2\pi{\sqrt{5}}})$ ?
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1answer
67 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 ...
1
vote
2answers
136 views

Rank these notations from most valid to least valid

$$\mathbb{Z}[\sqrt{-3}]$$ $$\mathbb{Z}\left[\frac{i \sqrt{3} - 1}{2}\right]$$ $${\bf{A}}(-3)$$ $$\mathbb{Z}[\omega]$$ $$\mathbb{Z}\left[-\frac{1}{2} + \frac{\sqrt{-3}}{2}\right]$$ ...
1
vote
0answers
22 views

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 ...
3
votes
1answer
137 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 ...
1
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1answer
54 views

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 $Q(\zeta^{(1/2)})$ ?
2
votes
1answer
82 views

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 ...
2
votes
1answer
52 views

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 ...
1
vote
1answer
211 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 ...
11
votes
5answers
1k views

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 ...
5
votes
4answers
168 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 + ...
1
vote
0answers
54 views

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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votes
0answers
33 views

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 ...
1
vote
1answer
35 views

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 ...
2
votes
1answer
35 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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votes
0answers
23 views

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 ...
5
votes
0answers
89 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 ...
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votes
0answers
26 views

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 ...
2
votes
1answer
38 views

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 ...
2
votes
0answers
32 views

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 ...
3
votes
1answer
43 views

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 ...