Questions related to the algebraic structure of algebraic integers

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Direct Proof of Divisibility in Extensions of Number Fields

Let $L/K/\mathbb Q$ be a tower of number fields. The result I want to show is that the discriminant $\Delta_K$ divides the discriminant $\Delta_L$. I was wondering if there was a "direct" proof of ...
2
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1answer
14 views

Discriminant of a product of polynomials

Let $f,g$ be irreducible, monic and in $\mathbb Z[x]$. Then (I hope this is correct) $disc(f\cdot g)=disc(f)\cdot disc(g)\cdot\prod_i\prod_j(a_i-b_j)^2$ where the $a_i$ are the roots of $f$ ...
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1answer
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Computing class group of $\mathbb Q(\sqrt{6})$

I am calculating the class group of $\mathbb Q(\sqrt 6)$. My working is as follows: The Minkowski bound is $\lambda(6)=\sqrt 6<3$ so we only need to look at prime ideals of norm $2$. $2$ divides ...
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1answer
15 views

If prime p doesn't divide the class number, then if I is an ideal of $O_K$, and $I ^{p}$ is principal, then I is principal

If a prime p doesn't divide the class number of a number field K, then if I is a non-zero ideal of $O_K$, and $I ^{p}$ is principal, then I is principal.
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1answer
31 views

Splitting Field of a real cubic

Let $f(X)$ be a cubic with 3 real roots, integer coefficients irreducible over $\mathbb{Q}$. Let $\alpha$ be one of these roots, and consider the number field $\mathbb{Q}(\alpha)$. Dirichlet's unit ...
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Is the discriminant of a polynomial surjective onto $\mathbb Z$?

Consider polynomials of degree two over $\mathbb Z$: $f = ax^2+bx+c$ The discriminant is $D = b^2-4ac$ And we can show that $D=2$ is not a possible value for $D$. I wonder if the value ...
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1answer
36 views

If $(3)=\mathfrak p_3\mathfrak p_3'$ then we can write $\mathfrak p_3=(3,1+\sqrt{17})$

Why if $(3)=\mathfrak p_3\mathfrak p_3'$ in $\mathbb Z[\sqrt{-17}]$ then we can write $\mathfrak p_3=(3,1+\sqrt{-17})$ I saw here in the first exercise that the author already knows how to ...
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1answer
30 views

A Question about Algebraic Integers

I need to prove a lemma, which uses the following fact: If $\alpha$ is an algebraic number of degree $m$ over $\mathbb{Q}$. Define $\mu(\alpha)$ to be max$\{ |\alpha_i| \}$, where $\alpha_1=\alpha$, ...
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1answer
21 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 ...
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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 $\mathbb Z[\omega]$ ...
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1answer
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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; ...
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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 ...
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0answers
42 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.
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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 $n = 2$ ...
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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 ...
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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 ...
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$x\in\mathcal{O}_K$ 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 ...
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12 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 : ...
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68 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 ...
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2answers
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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 $\phi$. ...
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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: ...
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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 ...
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1answer
42 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 ...
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1answer
12 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 ...
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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 ...
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96 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 ...
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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, ...
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1answer
33 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 ...
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2answers
88 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
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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 ...
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+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 ...
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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 $\mathbb Z[\omega]$? ...
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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 ...
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2answers
50 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 $\mathbb F_p[x]/(x^2)$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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}}, ...
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2answers
26 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 ...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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1answer
42 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 ...
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1answer
54 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 ...