Questions related to the algebraic structure of algebraic integers

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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 $\sigma_1,..\sigma_n$ ...
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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} ...
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1answer
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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 ...
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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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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 $\alpha$ ...
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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 ...
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3answers
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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 ...
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1answer
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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 $\mathfrak{P}$ of ...
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1answer
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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 ...
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1answer
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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 ...
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Mordell Equation $y^2 = x^3 - 20$. [on hold]

Prove that the only integral solutions to $y^2 = x^3 − 20$ are $(x, y) = (6, \pm14)$.
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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 ...
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1answer
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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 $\delta'$ ...
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2answers
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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 ...
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1answer
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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 ...
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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 ...
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1answer
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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
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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
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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
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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
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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
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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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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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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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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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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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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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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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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
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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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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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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
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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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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 ...
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1answer
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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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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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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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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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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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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 ...