Questions related to the algebraic structure of algebraic integers
3
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0answers
23 views
Strengthened Dirichlet's Unit Theorem for Cyclotomic Fields
If $p$ is an odd prime, $\xi$ is a $p$th root of unity, and $\mu_k = \frac{1-\xi^k}{1-\xi}$, then $\mu_2, \mu_3, \ldots, \mu_{\frac{p-1}{2}}$ are multiplicatively independent. I would greatly ...
0
votes
1answer
18 views
Quadratic fields with cyclic class group
Let $\mathbb{K}$ be a real quadratic field, with discriminant $d_{\mathbb{K}}<36$. Then the Minkowski's bound is $\frac{1}{2}\sqrt{d_{\mathbb{k}}}<3$. By the Minkowski's Theorem, each ideal ...
6
votes
1answer
172 views
The ring of integers of a number field is finitely generated.
For a number field $K$, we define the ring of integers of $K$ to be
$$\mathcal{O}_K:=\{x\in K\big|\ (\exists f\in\mathbb{Z}[X])(f\ \text{ is monic and } f(x)=0)\}.$$
Is there any easy way to see from ...
0
votes
1answer
113 views
Cyclotomic euclidean number fields
I´m here because I want to repeat my question about the norm-euclidean algorithm in a particluar cyclotomic integer ring.
Let $L=\mathbb{Q}(\zeta_{32})$ and $A=\mathbb{Z}[\zeta_{32}]$. On the page ...
0
votes
0answers
20 views
Hilbert class field
I got this doubt after going through tables given by Henri Cohen: Advanced topics in Computational number theory on "Hilbert Class Field of Imaginary quadratic field". pg no.539-542, sec 12.1 and a ...
1
vote
1answer
33 views
Cardinality of prime divisors in cyclotomic fields
1) For $p$ an odd prime, let $K_{n} = \mathbb{Q}[e^{\frac{2\pi i}{p^{n}}}] \ $ , and let $R_{n}$ be the ring of integers of $K_{n}$. Let $q\mathbb{Z} \ $ be a prime ideal of $\mathbb{Z} \ $, with ...
1
vote
1answer
52 views
A question on the Chinese Remainder Theorem
This is a question from Lang's ANT, Thm 2 (ch.7, $\S2$).
Let $k$ be a number field and $A$ its adele group.
In the proof, Lang states
Given $x\in A$, let $m$ be a rational integer such that ...
1
vote
1answer
27 views
How many ideals with fixed norm value
I'm doing some number theory exercises with solutions. Well, I can't understand the solution to one of them.... the exercise asks for how many ideals $A$ in $R$ we have $N(A)=2^2\cdot 11^3$, where $R$ ...
5
votes
1answer
31 views
Tate's Thesis: Meaning of Local Functional Equation
I am studying the development of Tate's Thesis in Lang's Algebraic Number Theory and have a conceptual question.
The setting: Let $k=\mathbb{Q}_p$. Let $\mu$ be the unique Haar measure giving ...
3
votes
2answers
33 views
The discriminant of a cubic extension
Let $K=\mathbb{Q}[\sqrt[3]{7}]$. I want to find the discriminant $d_K$ of the number field $K$.
I have computed $\operatorname{disc}(1,\sqrt[3]{7},\sqrt[3]{7^2})=-3^3\cdot 7^2$. I know that $d_K$ ...
2
votes
1answer
20 views
Norm computation in number fields
Let $\alpha:=\sqrt[3]{7}$ and let $K:=\mathbb{Q}[\sqrt[3]{7}]$. Consider a generic algebraic integer $a+b\alpha+c\alpha^2$, with $a,b,c\in\mathbb{Z}$. I want to find $N(a+b\alpha+c\alpha^2)$, where ...
4
votes
3answers
74 views
Ramification of primes without knowing the discriminant
Let $\mathbb{K} = \mathbb{Q}[\sqrt[3]{5}] \ $, and let $\mathbb{L}$ be the normal closure of $\mathbb{K}$.
Let $\mathbb{O}_{\mathbb{K}} \ $ be the integral closure of $\mathbb{Z}$ in $K$ and ...
1
vote
1answer
32 views
A set of prime factors of an integer in $\mathcal{O}_k$
I've got a basic question from Thm 2 (ch.7, $\S2$) of Lang's Algebraic Number Theory.
Let $k$ be a number field and $A$ its adele group. Let $S_{\infty}$ be the set of Archimedean absolute values of ...
2
votes
1answer
24 views
The fundamental unit in the ring of algebraic integers.
I can't understand my own notes in number theory....I've written: let $K$ be a number field, $\mathscr{O}_K$ its ring of integers and $U_K=\mathscr{O}_K^{\star}$ the multiplicative group of units in ...
0
votes
0answers
17 views
Hilbert polynomial of $Q(\sqrt{-14})$
The Hilbert Class Field of $K=Q(\sqrt{-14})$ is $L=K(\alpha)$, where $\alpha=\sqrt{2+\sqrt{2}+1}$. Then how does one obtain the Hilbert Class field of discriminant as $X^4-X^3+X+1$.
1
vote
1answer
50 views
Finite and infinite primes of a number field
Let $K$ be a number field. How to find the finite primes of a number field. can any one give some illustrations.
6
votes
2answers
192 views
How to compute a discriminant
Let $\alpha$ be a root of the irreducible cubic polynomial $x^{3}+px+q$, $p,q\in \mathbb{Q}$. How can I compute the discriminant $\Delta(1,\alpha,\alpha^{2})$ relative to $\mathbb{Q}(\alpha)$?
3
votes
2answers
21 views
On the absolute norm of an ideal
Let $K$ be a number field, with number ring $\mathscr{O}_K$. Let $\mathfrak{a}$ be an ideal in $\mathscr{O}_K$ and let $\mathfrak{N}({\mathfrak{a}})$ denote the absolute norm of $\mathfrak{a}$. How ...
-1
votes
0answers
28 views
Discrete Logarithm with map to p-adics
Given $h_1 \in H \leq G$ with $H = (h)$ discrete logarithm finds $k \in \Bbb Z$ such that $h^k=h_1$ and if $|H|=n$ is finite, then $k \in \Bbb Z / n\Bbb Z$ holds.
Can we have a workable definition of ...
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votes
0answers
32 views
P-adic Unitary and Unimodular matrices
Is there a concept of a matrix that is simultaneously Unitary and Unimodular Matrices over $GL_n(\Bbb Z_p)$ (padic)?
Can they have entries only in $\Bbb Z/(p)\Bbb Z$?
In the real and complex case ...
0
votes
0answers
10 views
Division in a complete subring of a local field
Suppose $A$ is a complete subring of a local field such that a prime element $\pi$ belongs to $A$. Is it true that if $\beta=\pi^k u$ (with $k\ge 0$ and $v(u)=0$) and $\beta\in A$ then also $u\in A$?
...
3
votes
0answers
39 views
A criterion for a ramification of an algebraic number field
Let $K$ be an algebraic number field.
Let $h$ be the class number of $K$.
Let $l$ be a prime number.
Suppose $l$ does not divide $h$.
Let $L/K$ be a Galois extension of degree $l$.
Let $H$ be the ...
3
votes
1answer
62 views
Ray class field - Ring class field
Let $K=Q(\sqrt{-d})$ be an imaginary quadratic field. Let $O_{K}$ be its maximal order and $O$
be any order of $K$. Let $m$ be the conductor. can the ray class field and ring class field be same?
3
votes
1answer
36 views
Norm of an element coprime to a prime algebraic number
Let $\pi:=1+\sqrt{3}$ be an element of $\mathbb{Z}[\sqrt{3}]$. I have proved that $\pi$ is a prime number in $\mathbb{Z}[\sqrt{3}]$. Now let $\alpha$ be another element in $\mathbb{Z}[\sqrt{3}]$, such ...
0
votes
0answers
38 views
$p=a^2-3b^2$ or $p=3b^2-a^2$, $p$ prime
Consider the number field $K=\mathbb{Q}[\sqrt{3}]$, with number ring $\mathbb{Z}[\sqrt{3}]$. Now, $3$ is a square modulo a prime $p$ iff the minimal polynomial $x^2-3$ of $\sqrt{3}$ splits in ...
0
votes
1answer
31 views
Prove that the subdomain $\mathbb{Z}+ 7\mathbb{Z}\sqrt{2}$ of the Euclidean domain $\mathbb{Z}+\mathbb{Z}\sqrt{2}$ is not Euclidean
I need help to prove that the subdomain $\mathbb{Z}+ 7\mathbb{Z}\sqrt{2}$ of the Euclidean domain $\mathbb{Z}+\mathbb{Z}\sqrt{2}$ is not Euclidean.
I have been using the Alaca & Williams book, ...
8
votes
2answers
381 views
Value of cyclotomic polynomial evaluated at 1
Let $\Phi_n$ be the usual cyclotomic polynomial (minimal polynomial over the rationals for a primitive nth root of unity).
There are many well-known properties, such as $x^n-1 = \Pi_{d|n}\Phi_d$.
...
1
vote
1answer
17 views
Lemma for the construction of the reciprocity map
I do not understand the highlighted part in the following proof, namley that $N(\tilde x)=1$.
To give some context, this proof is taken from Neukirch's Algebraic Number Theory, where $\tilde K$ ...
0
votes
1answer
23 views
On the ring of integers of cyclotomic fields
I'm doing an exercise whose main purpose is to show that $\mathscr{O}_K=\mathbb{Z}[\xi]$, where $K=\mathbb{Q}[\xi]$ and $\xi$ is a primitive $p$-th root of $1$, $p$ prime.
So, let ...
1
vote
1answer
41 views
On a proof of a number field theorem
$\textbf{Theorem}\ $For every integer $n\geq 2$, let $\mathbb{Q}[\zeta_n]$ be the cyclotomic field generated by the primitive $n$-th roots of $1$. Suppose $\mathbb{Q}[\zeta_m]=\mathbb{Q}[\zeta_n]$ ...
