Questions related to the algebraic structure of algebraic integers

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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
127 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 ...
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0answers
27 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 ...
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1answer
42 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
296 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 ...
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2answers
62 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
69 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
69 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 ...
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0answers
30 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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1answer
72 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
23 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) ...
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3answers
91 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
74 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
38 views

Rogers-Ramanujan Continued Fraction

How to calculate Rogers-Ramanujan Continued Fraction $R(e^{-2\pi{\sqrt{5}}})$ ?
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1answer
59 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
136 views

Rank these notations from most valid to least valid

$$\mathbb{Z}[\sqrt{-3}]$$ $$\mathbb{Z}\left[\frac{i \sqrt{3} - 1}{2}\right]$$ $${\bf{A}}(-3)$$ $$\mathbb{Z}[\omega]$$ $$\mathbb{Z}\left[-\frac{1}{2} + \frac{\sqrt{-3}}{2}\right]$$ ...
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0answers
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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
118 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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0answers
41 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 $Q(\zeta^{(1/2)})$ ?
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1answer
78 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
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1answer
49 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 ...
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1answer
201 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
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 ...
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4answers
156 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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0answers
52 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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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
31 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 ...
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1answer
33 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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0answers
21 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 ...
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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
36 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 ...
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0answers
30 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 ...
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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
155 views

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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0answers
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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
66 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}) ...
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2answers
108 views

The prime elements of the ring $ \mathbb{Z} \! \left[ \dfrac{i \sqrt{3} - 1}{2} \right] $.

I have to study the prime elements of the ring $ \mathbb{Z} \! \left[ \dfrac{i \sqrt{3} - 1}{2} \right] $. For the moment, I cannot find the general form of such elements. Can you help me? Thanks! :) ...
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2answers
64 views

Does the data of Galois group, ramified places, and inertia groups, determine a Galois number field?

Suppose I tell you that $K/\mathbb{Q}$ is a finite Galois extension, and I specify the Galois group $G$, and suppose further that I give you a finite list $S$ of places of $\mathbb{Q}$ and for each ...
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1answer
27 views

Associated order of a Galois extension of number fields

I have a finite Galois extension $L/K$ of number fields with group $G$. Let $\mathcal O_L$ be the ring of algebraic integers of $L$. We let $\mathcal A_{L/K}$ be the subring $\{x\in K[G]:x\mathcal ...
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1answer
41 views

Polynomial Diophantine Equations

So in general how does one decide if: $$ a_0 + a_1x + a_2x^2 ... a_{n_1}x^{n_1} = b_1y + b_2y^2 ... + b_{n_2}y^{n_k}$$ Has solutions for integers $x,y$ given real numbers $a_0, a_1. .. a_{n_1}, b_1 , ...
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1answer
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Is there a simple procedure to produce algebraic numbers of modulus one that are not roots of unity?

Let $z=e^{i\theta}$ be a complex number of modulus $1$. Trivially if $z$ is a root of unity then $z$ is also an algebraic number, but the converse is known to be false : $z$ can be algebraic without ...
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2answers
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Show that it doesn't exist any of natural number $ n = 4m + 3$ that $ n= x^2+y^2 $ for any natural x and y [duplicate]

Show that it doesn't exist any of natural number $ n = 4m + 3$ that $ n= x^2+y^2 $ for any natural x and y Show that every prime number in form $ p=4m+1 $ could be showed as $ p = x^2+y^2$ (x and y ...
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2answers
53 views

Is $(x^2 + 1, y^2 + 1)$ a prime ideal in $\mathbb{Q}[x, y]$?

At first I was looking for a ring homomorphism from $\mathbb{Q}[x, y]$ to a domain with $(x^2 + 1, y^2 + 1)$ as it's kernel, but I could not find one. Now I am thinking: maybe $(x + y)(x - y) = x^2 ...
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1answer
240 views

Why is $(\sqrt{2}+\sqrt{3})^{2008}$ so close to an integer?

Using 5000-digit precision in PARI/GP, I discovered that the fractional part of $(\sqrt{2}+\sqrt{3})^{2008}$ is extremely small, less than $10^{-999}$. Is there a simple explanation for this fact ? ...
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1answer
37 views

ideal calculations: $2\mathcal{O}_K=\mathfrak{B}^4$ in the ring of integers of $K=\mathbb{Q}(i,\sqrt{2m})$

Let $K=\mathbb{Q}(i,\sqrt{2m})$ where $m \in \mathbb{Z}$ is odd and squarefree. Let $\alpha = (1+i)\sqrt{2m}/2$. Then $\alpha^2=im$, such that $\alpha$ is part of the ring of integers $\mathcal{O}_K$. ...
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1answer
28 views

Ideal in Dedekind domain

Let $D$ be Dedekind domain and $I$ nonempty ideal in $D$. I have to show that there are only finitely many ideals $J$ in $D$ such that $I$ is contained in $J$. My first idea would be: assume that ...
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0answers
40 views

divisibility question involving primes

I have a question concerning the following divisibility problem. For any prime $p$ we define set: $\mathtt{V_{p}}:=\Biggl\{F\in\Phi\Biggl|\begin{cases}p^2\nmid ...
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1answer
51 views

Before we consider the prime decomposition

Let $L/K$ be a number field extension. Let $I$ be a prime ideal of $O_K$. How to prove that $IO_L\neq O_L$? It looks there should be a very fast way to see this, but I don't know how.
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2answers
60 views

Transcendentals as the Roots of Infinite Polynomials

I have always been taught that the difference between an algebraic and a transcendental number is that the former is the root to a polynomial of ${\bf finite}$ degree with integer coefficients. I did ...