Questions related to the algebraic structure of algebraic integers

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2answers
51 views

Ring homomorphism takes discriminant to discriminant

Let $R[x] \xrightarrow{\sigma} S[x]$ be a ring homomorphism where $R,S$ are integral domains of characteristic $0$. Is it true that for any monic polynomial $f(x) \in ...
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1answer
26 views

If the limit of a sequence of algebraic integers is algebraic, does it need to be an algebraic integer?

Consider a sequence $\{\alpha_n\}$ of algebraic integers and let $\alpha = \lim_{n \to \infty} \alpha_n$, where the limit is taken with respect to the usual absolute value in $\mathbb{C}$, and suppose ...
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1answer
26 views

Finding powers of prime ideals from its generators and understanding generator notation

I am trying to understand ideal notation with pointed brackets and how to use it. For instance, if I had an ideal $\mathfrak{a}=\left<2,1+\sqrt{-5}\right>$, where $2$ and $1+\sqrt{-5}$ are its ...
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1answer
28 views

Norm of an ideal is finite

I want to show that the norm $N_{K/\mathbb Q}(\mathfrak{a})$ of $\mathfrak{a}$ a nonzero integral ideal of a number field $K$ is finite, and so $N_{K/\mathbb Q}(\mathfrak{ab})=N_{K/\mathbb ...
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1answer
29 views

Writing a Gauss sum as a sum over divisors

Let $\chi$ be a Dirichlet character modulo $q$ induced by a primitive character $\chi^*$ modulo $d$ for some divisor $d$ of $q$. Let $n$ be a positive integer, and consider the generalised Gauss sum ...
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3answers
22 views

Determinant of a Vandermonde matrix of roots of monic polynomial with integer coefficients

Let $p(x)=\sum_{i=1}^n a_ix^i$ with $a_i$ an integer for all $i$ and $a_n=1$ such that $p(x)$ has only real roots, and let $\lambda_1,\ldots,\lambda_n$ be the $n$ roots of this polynomial. Then the ...
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0answers
38 views

Elementary solution to the Mordell equation $y^2=x^3+9$?

I've recently been wondering how to solve the equation of mordell for k=9, namely: (y^2=x^3+9). It reduced to solving the Thue equation (|a^2-2b^3|=3).Interestingly, the equation has several ...
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0answers
39 views

Ramanujan conjecture and Langlands program

In the article http://www.thehindu.com/sci-tech/science/the-legacy-of-srinivasa-ramanujan/article2746988.ece, it was mentioned that "This conjecture, later called Ramanujan's conjecture, came to ...
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5answers
152 views

Why is quadratic integer ring defined in that way?

Quadratic integer ring $\mathcal{O}$ is defined by \begin{equation} \mathcal{O}=\begin{cases} \mathbb{Z}[\sqrt{D}] & \text{if}\ D=2,3\ \pmod 4\\ ...
4
votes
1answer
52 views

Number of finite extensions of $p$-adic number field of given degree $n$

Let $p$ be a prime number, $\mathbb{Q}_p$ the $p$-adic number field. We fix an algebraic closure $\Omega$ of $\mathbb{Q}_p$. Any algebraic extension of $\mathbb{Q}_p$ is assumed to be a subfield of ...
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1answer
55 views

Motivation for the definition of the Artin Conductor of a representation

I'm trying to figure out what the Artin Conductor of a representation is using Chapter IV.$2\text-3$ of Serre's Local Fields, and I'm struggling to understand the motivation behind its definition. ...
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1answer
35 views

proving Fermat's theorem on $p = x^2 + 3y^2$

Here is a modern proof from the notes primes presented by quadratic forms. We are interested in $p = x^2 + 3y^2$ so we would like to have something like: $$ p = (x + y\sqrt{-3})(x - y\sqrt{-3}) = ...
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1answer
39 views

How many positive solutions are there (Positive 3 tuples)?

I want to find how many positive solutions for the Diophantine equation $4x + 2y + 5z = 100$ I found a particular solution $(x,y,z) = (50,-50,0)$ then I found a general solution (basis) $s(-2,-1,2) + ...
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1answer
9 views

prime ideal of integral closure on the decomposition iff lies above the prime ideal of the ring

I'm having troubles proving the following proposition. In every reference I read, they mark this proposition as "clear" or "trivial", but I am unable to prove it. Some help? Let $A$ be a Dedekind ...
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0answers
12 views

Number of isotropic vectors of a hermitian form

Good evening, I have a question about isotropic vectors in hermitian spaces and I hope someone can help me out. Let K be a local non-dyadic field and $\pi$ a prime element (so 2 is a unit). Let $h$ ...
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1answer
19 views

Properties of the exponent function attached to a nonzero prime ideal in a Dedekind domain

I want to prove properties of $v_\mathfrak{p}$, which I have been told is: "the exponent function attached to a nonzero prime ideal $\mathfrak{p}$ that maps a given nonzero fractional ideal to the ...
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0answers
26 views

equivalence of norms vs equivalence of absolute values

Consider a field, $k$. A norm on $k$ is a map, $|-| : k \to \mathbb{R}$ such that $(1)$ $|v|= \iff v=0$ $(2)$ $|vw| = |v||w|$ $(3)$ $|v+w| \leq |v|+|w|$ and we say two norms $|-|_1, |-|_2$ are ...
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1answer
45 views

$(1-\zeta_m)$ is a unit in $\mathbb{Z}[\zeta_m]$ if m contains at least two prime factors

We know that for $m=p^r, 1-\zeta_m$ is a prime.Now suppose that m has at least 2 distinct primes appearing in its prime factorization,we need to show that $1-\zeta_m$ is a unit in its ring of integers ...
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votes
2answers
88 views

Why is $(3,1+\sqrt{-5})^2=(2-\sqrt{-5})$ in $\mathbb{Z}[\sqrt{-5}]$

I want to show $(3,1+\sqrt{-5})^2=(2-\sqrt{-5})$ in $\mathbb{Z}[\sqrt{-5}]$. It's easy to see $(2-\sqrt{-5})\subset (3,1+\sqrt{-5})^2$ since $2-\sqrt{-5}=3-(1+\sqrt{-5})$ is in $(3,1+\sqrt{-5})^2$. ...
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1answer
43 views

P-adic valuation for ideals

Let $A$ be a Dedekind domain and $\mathfrak{a},\mathfrak{b}$ be fractional ideals of $A$. Then we know that $\mathfrak{a}$ and $\mathfrak{b}$ can be decomposed into ...
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1answer
36 views

Describing the Inertia group of a number field

Let $ K \subseteq L$ be number fields and $\pi$ be a prime ideal of $L$. $G = \operatorname{Gal}\left(L/K\right)$ Let $D = \{\sigma \in G\:|\:\sigma(\pi) = \pi\} $ be the decomposition group for ...
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3answers
100 views

Showing these prime ideals are principal

Let $K=\mathbb{Q}(\theta)$ be a number field where $\theta$ has minimal polynomial $x^3-9x-6$. I had to factorise the ideals $(2)$ and $(3)$ into prime ideals, for which I got $(2) = ...
5
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2answers
61 views

Relation between units in $\mathbb{Q}(\sqrt{13})$ and integral solutions to $x^2 - 13y^2 = \pm 1$

I have shown that in the number ring of $\mathbb{Q}(\sqrt{13})$, the units are precisely $\pm \left(\frac{3+\sqrt{13}}{2} \right)^n$. How can one deduce the integral solutions to the related Pell's ...
4
votes
1answer
54 views

Intermediate fieds for a Cyclotomic Polynomial of order $27$?

I would like to determine the Galois structure of the field $K=\Bbb Q(\zeta_{27})$--the rationals adjoined a primitive $27^{th}$ root of unity. That is to say I would like to determine the ...
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2answers
63 views

A break in symmetry between Algebraic number fields over Q and otherwise

Most of the theorems in algebraic number theory seem to generalize to arbitrary base fields apart from $\mathbb{Q}$ apart from one. The characteristic of the residue field is equal to positive prime ...
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0answers
64 views

A question about number theory( ask for directions)

I have a problem in mind and I want to know which research field it belongs to, then I can read something more specific(maybe). The problem is : consider two sequences of integers $s_n,r_n$ then ...
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1answer
21 views

Existence of Conductor for Cyclotomic Extension (pg 200, Serge Lang A.N.T.)

Let $\zeta$ be a primitive $m$th root of unity, $m \not\equiv 2 \pmod 4$, and $K = \mathbb{Q}(\zeta)$. For $p$ prime and unramified i.e. $(m,p) = 1$, I know that the Artin symbol $(p, K/\mathbb{Q})$ ...
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3answers
67 views

Finding units in cyclotomic fields

I want to classify the six units in $\mathbb Z[\zeta_3]$, where $\zeta_3$ is a primitive cube root of unity. I know the basic idea of this is to show that the norm of $\alpha \in \mathbb Z[\zeta_3]$ ...
4
votes
1answer
44 views

Intersection of ring and prime ideal

Give an example of an extension $B/A$ of rings, with $B$ an integral domain and a nonzero prime ideal $\mathfrak{p}$ of B such that $\mathfrak{p} \cap A=(0).$ I don't know where to begin with this.. ...
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2answers
33 views

Finding the minimal polynomial and its conjugates without a matrix

Let $K=\mathbb Q\left(^3\sqrt{5}\right)$ and $\alpha=a+b\left(^3\sqrt{5}\right)+c\left(^3\sqrt{5}\right)^2$. How do I find the minimal polynomial $f_\alpha$ of $\alpha$ over $\mathbb Q$? I am already ...
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0answers
28 views

Exercise to determine $\mathcal{O}_K$

Exercise: Let $p$ and $q$ be prime with $p \equiv 1 \bmod{4}$ and $ q \equiv 3 \bmod{4}$. Let $K=\mathbb{Q}(\sqrt{p},\sqrt{q})$, determine $\mathcal{O}_K$. I'm stuck with this exercise. As a ...
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2answers
48 views

Requirement on Norm for units in Cyclotomic Fields

Consider $\zeta_p$ a primitive $p^{th}$ root of unity. Prove that $\alpha \in \mathbb Z[\zeta_p]$ is a unit iff $N(\alpha)=\pm1$. I'm not even sure where to start with this. I know that ...
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0answers
40 views

Finding Norm, Trace and Characteristic Polynomial of field extensions

If $K=\mathbb Q(\beta)$ with $^3\sqrt{5}$ and $\alpha=a+b\beta +c\beta^2$, I want to find $N_{K/\mathbb Q}(\alpha), Tr_{K/\mathbb Q}(\alpha)$ and $\chi_{K/\mathbb Q}$ of $\alpha$ for in two different ...
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1answer
41 views

$\mathcal{O}_K$ analogous to $\mathbb{Z}$?

The definition of $\mathcal{O}_K$ isn't very well explained or motivated in my textbook. Let $K$ be a field a field with $\mathbb{Q} \subset K \subset \mathbb{C}$. $\mathcal{O}_K$ consists of all ...
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votes
1answer
27 views

Basis for number fields and rings of integers

The notation of algebraic number theory is frustrating for me. What does the notation $\mathbb{Q}(\alpha)$ mean? Assuming $\alpha \notin \mathbb{Q}$, is it simply $\lbrace a+b\alpha|a,b\in \mathbb{Q} ...
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0answers
15 views

Under what conditions is a tower of quadratic extensions a UFD, GCD domain, or just an Integral Domain?

I have been studying towers of quadratic extensions to $\mathbb Q$ and have noticed the following: $\mathbb Q[\sqrt 2]$ and $\mathbb Q[\sqrt 2][\sqrt 3]$ are unique factorization domains(UFDs), but ...
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0answers
39 views

Show that $\sum_{k=1}^\infty 10^{-k!}$ is not algebraic

Exercise: Show that the number $\sum_{k=1}^\infty 10^{-k!} = 0.110001000\ldots$ is not an algebraic over $\mathbb{Q}$. I don't know how to start, can you give me any hints? Thank you!
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0answers
37 views

Norm in the cyclotomic integers

Let $\alpha \in \mathbb{Z}[\zeta_3]$, where $\zeta_3=e^{2\pi i/3}$ is a cube root of unity. So $\alpha=x+y\zeta_3$ for $x,y\in\mathbb{Z}$. Show that the norm $N(\alpha)$ can be written as ...
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0answers
26 views

Abelian extension of an algebraic number field whose Galois group is isomorphic to a given abelian group

Let $K$ be an algebraic number field, i.e. a finite extension of $\mathbb{Q}$. Let $G$ be a finite abelian group. Does there exist a Galois extension of $K$ whose Galois group is isomorphic to $G$? I ...
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1answer
42 views

Ideals in the ring of integers $\mathcal{O}_K$

I have two question about the proof of this theorem: Theorem: Let $K$ be an algebraic number field and $n = \dim_{\mathbb{Q}} K$. Then any ideal $I\neq 0$ in the ring of integers $\mathcal{O}_K$ ...
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1answer
43 views

Infinite subset with pairwise comprime elements

I'm looking for solution for this problem Let $a$ be an integer greater then 1. Then, the set $A\mathrel{:=}\{a^n(a+1)-1\mid n\in\mathbb{Z}_{+}\}$ a has an infinite subset b such that it's ...
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0answers
24 views

Classic proof that $N_{L/K}(P) = p^f$

Let $L/K$ be a finite extension of an algebraic number field $K$. Let $J$ be an ideal of $L$. Hilbert defined the relative norm $N_{L/K}(J)$ of $J$ as $N_{L/K}(J) = \Pi \sigma(J)$, where $\sigma$ ...
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0answers
26 views

Finding a basis for a cubic field

Let $K=\mathbb{Q}(\sqrt[3]{5})=\mathbb{Q}(\beta)$, and $\alpha=a+b\beta+c\beta^2\in K$. That is, $a,b,c \in \mathbb{Q}$. Compute $N_{K/\mathbb{Q}}(\alpha)$, $Tr_{K/\mathbb{Q}}(\alpha)$, and ...
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1answer
20 views

Cauchy sequence in a valuation ring

From Janusz's book algebraic number fields, chapter 2. Let $K$ be a complete field with respect to a non-Archimedean absolute value $|\cdot|$. Let $R$ be its valuation ring with maximal ideal ...
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votes
1answer
60 views

The maximal unramified extension of a local field may not be complete

While reading my notes of a course in local class field theory, I arrived to a remark where it is said that given a complete discrete valuation field $K$, its maximal unramified extension $$K^{ur}= ...
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1answer
20 views

Isomorphism between polynomial with indeterminate and integral element

If $x$ is an indeterminate, $A$ is a ring and $\alpha$ is in some ring containing $A$, is $A[x]$ is isomorphic to $A[\alpha]$ if $\alpha$ is integral over $A$? What about if $\alpha$ is transcendental ...
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1answer
38 views

prime ideal is a discrete valuation ring

I'm trying to show the following: Let $R$ be a subring of an integer ring $\mathcal{O}_K$, and suppose that as groups, $R$ is an index $m$ subgroup of $\mathcal{O}_k$. Prove that if $P \subset R$ is ...
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1answer
91 views

Fermat's Theorem on $p = a^2 + b^2$

I have read that Fermat predicted that for an odd prime $p$, $p = a^2 + b^2$ iff $p = 1 \pmod 4$. I heard that such a criterion could be possible for a given integer $n$ like $p = a^2 + n b^2$ ...
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0answers
29 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 ...
2
votes
1answer
42 views

Greatest common divisor problem on number theory. [duplicate]

Prove that if $\gcd(x, y) = 1$ then $\gcd(x + y, x - y) = 1$ or 2. I know that any linear combination of $x, y$ is multiple of 1 since $\gcd(x, y) = 1$ then the set would be $\{1, 2, 3, 4, 5, \ldots, ...