Questions related to the algebraic structure of algebraic integers

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2
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1answer
13 views

How does $ \text{Gal}(K / k) $ act on ideles?

Let $K/k$ be cyclic of degree $N$, Galois group $G$. I want to define some action of $G$ on the group of ideles $J_K$ which commutes with multiplication. A natural way to do this is to take each ...
0
votes
0answers
8 views

Reducing mod $n$ in $\mathcal O_{\mathbb Q (\zeta_n)}$

I noticed (see this question) that $\mathbb Z[\zeta_n]\cong \mathbb Z[\omega]$ and $\mathbb Z[i]$ when $n=3,4$ respectively and $\zeta_n$ is the primitive $n$th root of unity. Here $\mathbb Z[\omega]$ ...
0
votes
1answer
18 views

Two questions concerning ideal factorization and norm

$\bullet$ In $\mathbb Z[\sqrt{-5}]$ why is $(2)=(2,1+\sqrt{-5})(2,1-\sqrt{-5})$ Actually both ideals on the RHS contain $(2)$, but also their product ? Can we just multiply RHS in the normal sense; ...
0
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0answers
18 views

If $a \in \mathbb{A}$\{$\mathbb{0,1}$}, $b \in \mathbb{A}$\ $\mathbb{Q}$ then $a^b$ is transcedental.

For my math study, I have to prove the following: Let's denote the set of algebraic numbers with $\mathbb{A}$. Prove: If $a \in \mathbb{A}$\{$\mathbb{0,1}$}, $b \in \mathbb{A}$\ $\mathbb{Q}$ then ...
1
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0answers
38 views

Class number of $\mathbb Q(\sqrt{10}) $

I am interested in knowing how to compute the class number of $\mathbb Q(\sqrt{10}) $. I am confused with these class number computations.
4
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1answer
35 views

is there a negative integer which is a quadratic residue mod every prime $p\equiv 7\mod 8$

Is there a negative integer $n < 0$ such that the congruence $x^2 = n\mod p$ is solvable for every prime $p\equiv 7\mod 8$? If we remove the negativity condition it's well known that $n = 2$ ...
7
votes
0answers
67 views

Proving that the number of integer solutions of $x^2-Ny^2=1$ is infinite

I am trying to prove that the number of integer solutions of $x^2-Ny^2=1$ is infinite whenever N is a squarefree integer. For this I define norm of $a+b\sqrt N=a^2-Nb^2$. Now I prove that $a+b \sqrt ...
0
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0answers
18 views

explicitly splitting the hamilton quaternions over local fields

For simplicity, lets first consider the hamilton quaternions $$ H = \left(\frac{-1,-1}{\mathbb{Q}}\right)$$ This is the central division algebra over $\mathbb{Q}$ with $\mathbb{Q}$-basis given by ...
1
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0answers
13 views

$x\in\mathcal{O}_K$ can be written as product of irreducible elements

Lemma: Every element $x\neq 0$ of $\mathcal{O}_K$ with ($K$ an arbitrary number field) can be written as a product of irreducible elements. Proof: We prove this lemma by complete induction on ...
0
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0answers
11 views

How does class field theory help us deduce the splitting of nonprincipal prime ideals?

I had a general question about the significance of global class field theory. One of the goals, as I understand, is to answer the following question: Given $L/K$ abelian, $g$ a divisor of $[L : ...
2
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0answers
61 views

Class Group of $\mathbb Q(\sqrt{-15})$

Class Group of $\mathbb Q(\sqrt{-15})$ I used this paper for my attempt. First the discriminant of $\mathbb Q(\sqrt{-15})$ is the discriminant of the monic minimal polynomial of ...
1
vote
2answers
23 views

Image of archimedean place of a number field in $\mathbb C$

Let $L/K$ be a finite Galois extension of number fields and let $\phi$ be an embedding of $K$ into $\mathbb C$. Let $\psi_1$ and $\psi_2$ be two embeddings $L\to \mathbb C$ which extend $\phi$. ...
1
vote
2answers
30 views

Proof of equivalence of definitions of split primes etc.

I think my definitions of a prime being ramified, split and inert are non-standard. Also I do not see how my definitions are equivalent to (what appear to be) the standard ones. My definition: ...
2
votes
0answers
15 views

norm map and local class field theory

Let $K$ be a local field, say a finite extension of $\mathbb{Q}_p$ (which is the purpose of my interest). Let $L$ be an unramified extension of $K$. Local class field theory asserts that there ...
2
votes
1answer
41 views

Is $\mathbb{Z}(\sqrt[3]{5})$ a PID? Factorisation of the ideal $(2)$

I am given that for the ring of integers of $K = \mathbb{Q}(\sqrt[3]{5})$ is $\mathcal{O}_K = \mathbb{Z}(\sqrt[3]{5})$. I am supposed to factorise the ideals $(2), (3), (5)$ and $(7)$, show that all ...
0
votes
1answer
11 views

Ideal factorization Theorem, more generally

Consider Theorem 4.3.1 in link (it's quite long, so please open the pdf) I'm wondering if we can assume that the prime ideal we want to decompose is not $(p)$ with $p$ a prime in $\mathbb Q$, but a ...
0
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0answers
10 views

Function fields isomorphism

Determining if two number fields are isomorphic is a hard problem (Cohen, A course in computational algebraic number theory). Is determining if two functional fields are isomorphic a hard problem? Is ...
5
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0answers
91 views

A problem about $e^{2\pi i \alpha_1}+e^{2\pi i \alpha_2}+\cdots+e^{2\pi i \alpha_N}=0$

Let $\alpha_i\in [0,1),\; i\in \{1,\cdots,N\}$ for some positive integer $N$, such that $$e^{2\pi i \alpha_1}+e^{2\pi i \alpha_2}+\cdots+e^{2\pi i \alpha_N}=0$$ and if for any non-empty proper subset ...
11
votes
0answers
35 views

Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$.

We consider an analogue of the Dirichlet $L$-function in $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, ...
1
vote
1answer
32 views

Quotient of the ring of integers of a quadratic field by the ideal generated by a split integer prime.

I am wondering about primes $p$ in $\mathbb Z$ that are split in $\mathcal O_{K}$, $K=\mathbb Q(\sqrt d)$. Let $\omega=\sqrt d$ if $d \equiv 2,3 \mod 4$ and $\omega=\frac{1+\sqrt d}{2}$ if $d \equiv 1 ...
8
votes
2answers
87 views

How would you explain a quadratic field to a beginner?

How would you explain a quadratic field to a beginner? Eg. how did the subject first start? All the modern stuff they use to explain it makes it really confusing how one should think about it in more ...
2
votes
1answer
25 views

How do we determine the decomposition of $p\mathcal{O}_K$ in $K = \mathbb{Q}(\sqrt[3]{5})$?

Let $K = \mathbb{Q}(\sqrt[3]{5})$, and $\mathcal{O}_K$ be its ring of integers. In general, how do we decide the decomposition of $p\mathcal{O}_K$, for an odd prime $p$? I know that by Kummer's ...
6
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0answers
42 views
+250

Enumerating Bianchi circles

Background: Katherine Stange describes Schmidt arrangements in "Visualising the arithmetic of imaginary quadratic fields", arXiv:1410.0417. Given an imaginary quadratic field $K$, we study the Bianchi ...
1
vote
1answer
28 views

Are the Eisenstein integers the ring of integers of some algebraic number field? Can this be generalised?

The Eisenstein integers are $\mathbb Z[\omega]$ where $\omega$ is the primitive third root of unity. If $K$ is some algebraic number field, can $\mathcal O_K$ be isomorphic to $\mathbb Z[\omega]$? ...
0
votes
2answers
22 views

Extension of Completions of Number Fields

On p. 116 of Milne's notes on Algebraic Number Theory, he gives the following construction. Let $K$ be a field with a valuation $|\cdot|$ (archimedean or discrete nonarchimedean), and let $L$ be a ...
5
votes
2answers
49 views

Uniqueness of prime ideals of $\mathbb F_p[x]/(x^2)$

What are the prime ideals of $\mathbb F_p[x]/(x^2)$? I have been told that the only one is $(x)$, but I would like a proof of this. I want to say that a prime ideal of $\mathbb F_p[x]/(x^2)$ ...
2
votes
0answers
37 views

Is $\mathbb{Z}_{(p)}$ a Dedekind ring?

Is $\mathbb{Z}_{(p)}$ a Dedekind ring? (for a prime number $p$) By $\mathbb{Z}_{(p)}$ i mean the localization of $\mathbb{Z}$ at $p$. I know that one must check a couple of conditions, like it being ...
1
vote
1answer
31 views

prime ideal of $\mathcal O_K$ splits completely in tower of Galois extensions

The following exercise is taken from D.A.Cox's book "Primes of the form $x^2 +ny^2...$" He uses this in order to prove some intermediate steps before the proof of the main theorem in chapter 5, which ...
6
votes
1answer
39 views

How to write $\sqrt{4x^2 - 3}$ in the ring $\mathbb{Q}[x]/(x^3 - x - 1)$?

Consider the irreducible cubic equation $x^3 - x - 1 = 0$ and suppose we one of the roots $x$. The other two are $a,b$ such that $x + a + b = 0$ and $xab = 1$. Then $a$ and $b$ satisfy a quadratic ...
0
votes
0answers
18 views

Number of irreducible factors over $\mathbb{Q}_p[X]$

Let $p_1,\ldots,p_j$ be distinct primes and let $\beta=\sqrt{p_1}+\cdots+\sqrt{p_j}$. It's a fact that $$\mathbb{Q}(\sqrt{p_1},\ldots,\sqrt{p_j})=\mathbb{Q}(\beta)$$ and ...
1
vote
1answer
22 views

Quotient by power of maximal ideal

Suppose $R$ is a commutative ring (but see the edit portion below) and $\mathfrak{m}$ is a maximal ideal of $R$ such that $|R/\mathfrak{m}|<\infty$. Also assume that $k$ a positive integer. Is ...
0
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0answers
20 views

compute the grades over $\mathbb{Q}$ [duplicate]

Let $p_{1}$ $\neq$ $p_{2}$ $\neq$ $p_{3}$ prime numbers. Compute the grades over $\mathbb{Q}$ of the extension fields $\mathbb{Q} ( \sqrt{p_{1}}, \sqrt{p_{2}})$ and $\mathbb{Q} ( \sqrt{p_{2}}, ...
3
votes
2answers
25 views

Factorising ideals in the ring of integers of a quadratic field

In an undergraduate algebraic number theory course, I was given the question "If $K = \mathbb Q(\sqrt{-33})$ Factorise the ideal $(1+\sqrt{-33})\subset \mathcal O_K$ into a product of prime ideals." I ...
0
votes
1answer
22 views

definition of Artin map

Let $L/K$ be an unramified Abelian extension. Then the Artin symbol $ \left ( (L/K) / \mathfrak p \right) $ is defined for all prime ideals $\mathfrak p$ of $\mathcal O_K$. (because in an Abelian ...
3
votes
1answer
81 views

When $\cos x$ is transcendental?

About the transcendence of trigonometric functions I know that: 1) if $x$ is an algebraic number $\ne 0$ than $\cos x$ is transcendental. 2) if $p=\dfrac{m}{2^n}$ with $m,n \in \mathbb{Z}$ than ...
3
votes
1answer
28 views

Factor into primes in Dedekind domains that are not UFD's?

Does it make sense to factor numbers into prime numbers in Dedekind domains that are not unique factorization domains? I can't really see how it would make sense.
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0answers
53 views

A good book to read with Chapter III of Neukirch's “ANT”

The book Algebraic Number Theory from Neukirch is a beautiful book in ANT, but it still have a serious lack in examples and motivation to the concepts. I've already read the first two chapters of the ...
0
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0answers
25 views

$k(\mathfrak p)$ basis for $A / pA$

I'm reading this pdf which is showing that a rational prime $p$ ramifies if and only if it divides the discriminant of its number field $K$. I've come across the following line: Let $p \in \mathbb ...
0
votes
1answer
18 views

I need to show that if $K$ is of characteristic $0$,the algebra $A$ has a primitive generator.

Let $K$ be a field and $A$ a reduced K-algebra of finite dimension over $K$. I need to show that if $K$ is of characteristic $0$, $A$ has a primitive generator (i.e. $A=K[x], x \in A$) I've proved ...
2
votes
1answer
27 views

How to classify the rings of fractions of a principal ideal domain?

Let $A$ be a principal ideal domain and let $K$ be its field of fractions. I proved a) Every ring $B$ such that $A \subset B \subset K$ is a ring of fractions of $A$, and c) Show that any ring of ...
2
votes
1answer
41 views

The torsion subgroup of the group of units $R^{\times}$ is always equal to $\{\pm1\}$

In the ring of integers of the number field of degree $3$, the torsion subgroup of the group of units is always equal to $\{\pm1\}$ I found it here (Proposition $5.12$) only that the subgroup of ...
1
vote
1answer
53 views

Ideals in the ring of gaussian integers of a given norm

What are the ideals in the ring of gaussian integers of a given norm, (say $20$) ? The ring of integers is $\mathbb Z[i]$ and it is a PID, so any ideal must be principal. If the ideal $I$ is ...
2
votes
1answer
43 views

Which one is the ring of integers of $K$; $\mathbb Z[\alpha,\frac{\alpha^2}{2}]$ or $\mathbb Z[\alpha,\frac{\alpha+\alpha^2}{2}]$?

If $f := T^3 - T^2 + 2T + 8$ and $\alpha$ is the unique real root of $f$, If $K := \mathbb Q(\alpha)$, which one of the following rings are the ring of integers of $K$: $$\mathbb ...
1
vote
0answers
41 views

The Nagell-Ljunngren Equation

I have been trying to find the papers of Nagell and Ljunngren, which deal with the equation $$\frac{x^n - 1}{x - 1} = y^2$$ and solve it completely. Many papers cite these papers, but I haven't found ...
1
vote
0answers
15 views

Another real quadratic integer ring, Euclidean but not norm-Euclidean, with norm function needing only two adjustments?

I have come to know about $\mathcal O_K$ with $K = \mathbb Q(\sqrt{69})$. The norm function needs to be adjusted to absolute value, as is the case with other real rings, but it also needs to be ...
2
votes
0answers
24 views

On the existence of a polynomial in several variables with only one zero

Given an ordered field $\mathbb{K}$, then for all $n>1$ there exists $f\in\mathbb{K}[X_1,\ldots,X_n]\ s.t.\ \mathcal{V}^{\mathbb{K}^n}(f):=\{p\in\mathbb{K}^n:f(p)=0\}=\{(0,\ldots,0)\}$ For ...
0
votes
0answers
14 views

Show that the positive units of $K$ form a group isomorphic to $\mathbb Z$

Let $K$ be a cubic field such that $r_1=r_2=1$. Suppose $K$ is imbedded in $ \mathbb R$ ($r_1$ and $r_2$ are the usual notations, $r_1$ denotes the number of isomorphisms $\sigma : K \to \mathbb R$ ...
-4
votes
3answers
43 views

If α is algebraic over K then all the elements of K(α) are algebraic over K [closed]

Let $L$ : $K$ be a field extension, $\alpha \in L$ algebraic over $K$. Show that every element of $K(\alpha)$ is algebraic over $K$, where $K(\alpha)$ is the smallest subfield that contains $\alpha$ ...
1
vote
2answers
53 views

How to find the degree of an extension field?

How to find the degree of an extension field ? Let $f:=T^3-T^2+2T+8\in\mathbb Z[T]$ and $\alpha$ be the real root of $f$. Why is then $\mathbb Q(\alpha)$ is a number field of degree $3$ ? I've ...
1
vote
1answer
23 views

Norms and traces example

Example: Let $L=\mathbb{Q}(\sqrt{d})$ be a quadratic extention of $F=\mathbb{Q}$ with square-free integer $d$.Then, $g_{a+b\sqrt{d}}(X)=(X-a-b\sqrt{d})(X-a+b\sqrt{d})=X^2 -2aX+(a^2 -db^2),$ so, ...