Questions related to the algebraic structure of algebraic integers

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2
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2answers
26 views

behavior of a rational prime in quadratic extension (definition)

Let $ \mathbb Q \subset K=\mathbb Q (\sqrt{-n}) \subset L $, where $K/ \mathbb Q $ is a finite extension (i.e. $K$ is a number field) and $L/K$ is a maximal uramified abelian extension. If $p ...
5
votes
2answers
64 views

Solving polynomial equations using more than radicals

It is well known that one cannot solve every polynomial equation over $\Bbb Q$ using just radicals. In other words, let $A_n = \{x^n - a\mid a \in \Bbb Q\}$, $A = \cup_n A_n$ and $\bar{\Bbb Q}$ the ...
1
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1answer
25 views

For fractional ideal $I$ why is $I\cap R \supsetneq \{0\}$?

In the proof in my textbook that a fractional ideal $I$ in a quotient field $K$ of an integral domain $R$ has an inverse $$I^{-1} = \{ x\in K : x I \subseteq R\}\,,$$ it is used that there exists an ...
1
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0answers
33 views

Simple examples of fractional ideals

Let $K$ be the quotient field of an integral domain $R$. A fractional ideal $I$ is a subset of $K$ not $\{0\}$, for which a $0 \neq r \in R$ exists so that $r I \subseteq R$ is an ideal in $R$. ...
1
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0answers
24 views

Understanding linearly disjoint fields and how their rings of integers interact in a proof

Let $L,L'$ be linearly disjoint number fields (i.e. finite-degree extension of $\mathbb{Q}$). Their rings of integers are denoted $O_L,O_{L'}$. I am trying to understand a proof of how if $p$ is ...
4
votes
1answer
62 views

The splitting of an ideal

Let $K = \mathbb{Q}(\sqrt{-5})$. Now the ring of integers $\mathcal{O}_{K}$ is $\mathbb{Z}[i\sqrt{5}]$. I want to describe the ideal $(2)$ in $\mathbb{Z}[i\sqrt{5}]$ using the prime factorization. ...
4
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5answers
75 views

Show that $\dfrac{m - \sqrt[3]{2}}{\sqrt[3]{3}}$ is an algebraic integer.

Let $m$ be an integer such that $m \equiv 2 \pmod 3$. Show that the number $$\dfrac{m - \sqrt[3]{2}}{\sqrt[3]{3}}$$ is an algebraic integer. The usual technique, doing $x = \dfrac{m - ...
10
votes
3answers
159 views

Something screwy going on in $\mathbb Z[\sqrt{51}]$

In $\mathbb Z[\sqrt{6}]$, I can readily find that $(-1)(2 - \sqrt{6})(2 + \sqrt{6}) = 2$ and $(3 - \sqrt{6})(3 + \sqrt{6}) = 3$. It looks strange but it checks out. But when I try the same thing for ...
1
vote
1answer
80 views

The ideal $(p)$ always factors in the ring of integers of $\mathbb Q(\sqrt{2},\sqrt{3})$

Let $p$ be a prime integer. Is there a relatively elementary way to see that $(p)$ is never prime in the ring of integers of $\mathbb Q(\sqrt{2},\sqrt{3})$? One can prove this by looking at the ...
2
votes
0answers
33 views

The norm of an ideal and the norms of its elements

Let $F$ be a number field, $\mathfrak a$ a fractional ideal. If $\mathfrak a$ is a prime ideal in $\mathcal O_F$ lying over prime ideal $p\mathbb Z$ in $\mathbb Z$ then define its norm as $p^f$ where ...
0
votes
1answer
100 views

Prerequisites for Teichmuller Theory

What kind of prerequisites would be required for [Inter-Universal] Teichmuller theory or at least the closest generally known area near Mochizuki's work? (Starting from undergraduate math). I'd ...
0
votes
1answer
24 views

Why is the separable closure of a field in it's completion not complete.

I am currently reading "Algebraic Number Theory" by Neukirch and I am a bit confused by what is here. It is on page 143, it states this: $(K,v)$ is a nonarchimedian valued field and ...
1
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0answers
42 views

Show $\text{Gal}(K_\infty/\mathbb Q)\cong \mathbb Z_p^{\times}$

Let $\zeta_{p^n}$ be the primitive $p^n$-th root of unity where $p$ is a prime and $K_n=\mathbb Q(\zeta_{p^n})$ the $p^n$-th cyclotomic field. Let $K_\infty=\bigcup K_n$. Could someone give a proof ...
1
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3answers
41 views

Is it possible to factor $27$ in $\mathbb Z[\sqrt7]$ other than in $27=3\cdot3\cdot 3$?

I need to know if there exists an element of norm$27$ in $\mathbb Z[\sqrt 7]$, that is, whether there are any integer solutions to $a^2 - 7b^2 = 27$. When I used modular arithmetic, I find out that ...
0
votes
0answers
19 views

Volume of the lattice generated by an ideal

Let $F$ be a totally real number field, $\mathfrak a \subset F$ a fractional ideal. Consider a lattice in $\mathbb R^n$ consisting of vectors $(\sigma_1(v),..\sigma_n(v))$, where $\sigma_1,..\sigma_n$ ...
0
votes
1answer
47 views

field of rational functions of a curve

Let $C$ be the algebraic curve defined by the modular polynomial $\phi_N$ of order $N>1$ over the rational numbers, i.e. \begin{equation}C:=\text{specm}(\mathbb{Q}[X,Y]/\phi_N(X,Y)). \end{equation} ...
1
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1answer
56 views

Artin reciprocity theorem for Hilbert class field

In Cox's book "Primes of the form $x^2 + ny^2 $..." gives the following statement of Artin reciprocity theorem, for the Hilbert class field (i.e. maximal unramified Abelian extension) Artin's ...
0
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1answer
42 views

Prime decomposition in $\mathbb Z[x]/(x^3-x^2+x+1)$

If $K$ is the unique number field of discriminant $-44$, K is isomorphic to the field generated over $\mathbb Q$ by a root of the polynomial $x^3-x^2+x+1$ with $\mathcal O_K=\mathbb ...
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0answers
54 views

Show that $p$ is inert in the ring of integers of $K$

Let $p$ be a prime, $n$ a positive integer, $E$ a finite field with $p^n$ elements, and $\alpha ∈ E$ an element satisfying $\mathbb F_p(\alpha)=E$. If $\bar f$ is the minimal polynomial of $\alpha$ ...
0
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1answer
42 views

Primitive element theorem for finite fields

Primitive element theorem for finite fields Can you explain $2$ points in the proof of the proposition below $\bullet$ First $\alpha$ is the root of the polynomial $T^{p^s}-T$, because $\mathbb ...
3
votes
3answers
42 views

Split 16 Consecutive Integers into Two Subsets of 8 Integers

Show that any given set of sixteen consecutive integers {$x+1,x+2,\ldots,x+16$} can be divided into two eight element subsets with the properties that they have the same sum, the sums of the squares ...
3
votes
1answer
21 views

Presented matrix and number ring.

Let $V$ be the module generated by the column matrix $A= (2, 1+ \sqrt{-5})^T$. Prove that the residue of $A$ in $\mathbb{Z}[\sqrt{-5}]/ \mathfrak{P}$ has rank $1$ for every prime ideal ...
2
votes
1answer
27 views

Explicit calculation of residue field in Cyclotomic integers

I would like to show that $(1-\zeta)$ is a prime ideal in $\mathbb{Z}[\zeta]$, where $\zeta=\zeta_p =e^{\frac{2\pi i}{p}}$, for a prime $p$. I am aware that we can show $(1-\zeta)=(1-\zeta^i)$, for ...
1
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1answer
36 views

Find a counterexample to the following lemma if we change the statement slightly.

let K be an algebraic number field and let $O_K$ be its ring of integers. Lemma; Let $a,b$ be fractional ideals of $O_K$. If $b \subseteq a$ then there is an ideal $c$ such that $b=ac$. I need to ...
4
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0answers
79 views

Mordell Equation $y^2 = x^3 - 20$. [closed]

Prove that the only integral solutions to $y^2 = x^3 − 20$ are $(x, y) = (6, \pm14)$.
9
votes
4answers
95 views

Show that $\mathcal{O}_K$ is not UFD with $K = \mathbb{Q}(\sqrt{-13})$

Let $K = \mathbb{Q}(\sqrt{-13})$. Show that its ring of integers $\mathcal{O}_K$ is not an UFD. $-13 \equiv 3 \bmod{4}$, so $\mathcal{O}_K = \mathbb{Z}\bigl[\sqrt{-13}\bigr]$. We will use the ...
1
vote
1answer
34 views

Can someone prove or help me understand the following about Euclidean fields?

Why is it that if $\delta$ and $\delta'$ both divide $\alpha$ and $\beta$, and that every $\gamma$ which divides $\alpha$ and $\beta$ also divides $\delta$ and $\delta'$, then $\delta$ and $\delta'$ ...
6
votes
2answers
101 views

Is $3$ a prime element of $\mathbb{Z[\eta]}$?

How to check whether $3$ is a prime element or not in $\mathbb{Z[\eta]}$, where $\eta$ is a $17$th primitive root of unity. Also in general how can we check an element is prime or not in ...
3
votes
1answer
54 views

Frobenius automorphism and cyclotomic extension

There is a lemma from a lecture I attended where I scribbled down notes and try to make sense of the proof afterwards and there is a spot at which I am stuck. First I have to set up some notations and ...
3
votes
0answers
35 views

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
votes
1answer
22 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 $f$ ...
1
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1answer
59 views

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 ...
0
votes
1answer
24 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.
1
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1answer
39 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 ...
2
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0answers
45 views

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 ...
2
votes
1answer
41 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 ...
0
votes
1answer
31 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 $\alpha_1=\alpha$, ...
2
votes
1answer
34 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
21 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
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1answer
30 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
25 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 ...
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0answers
65 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
40 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 $n = 2$ ...
10
votes
0answers
107 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 ...
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0answers
20 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 ...
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0answers
17 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 ...
5
votes
1answer
120 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
82 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
33 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
3answers
44 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: ...