# Tagged Questions

Questions related to the algebraic structure of algebraic integers

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### Do the solutions to the unit equation lie dense in the complex numbers

Let $S\subset \overline{\mathbf{Q}}$ be the set of solutions to the unit equation, i.e., $S$ consists of algebraic integers $a$ such that $a$ and $1-a$ are units in the ring of algebraic integers. ...
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### Given $p \equiv q \equiv 1 \pmod 4$, $\left(\frac{p}{q}\right) = 1$, is $N(\eta) = 1$ possible?

Given distinct primes $p$ and $q$, both congruent to $1 \pmod 4$, such that $$\left(\frac{p}{q}\right) = 1$$ and obviously also $$\left(\frac{q}{p}\right) = 1$$ is it possible for the fundamental unit ...
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### On discriminants and nature of an equation's roots?

Edited: All equations in the post are assumed to have all real coefficients and are minimal polynomials. While trying to ascertain if the Brioschi quintic $B(x)=x^5-10cx^3+45c^2x-c^2=0$ could ever ...
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### Integer solutions of $x^3+y^3=z^3$ using methods of Algebraic Number Theory

I'm asked to prove that the famous equation $$x^3+y^3=z^3$$ has no integer (non-trivial) solutions, i.e. FLT for $n=3$ I'm aware that on this website there are solutions using methods of Number ...
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### What is the Strategy in Computing Ideal Class Number?

I found many examples on computing ideal class numbers, but none gave an explicit statement on what we are examining when we are running through a list of elements with their norms written out. The ...
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### Let $\alpha, \beta \in \mathbb{C}$, such that $\alpha + \beta$, and $\alpha\beta$ are algebraic. Show that $\alpha$ and $\beta$ are algebraic.

Let $\alpha, \beta \in \mathbb{C}$, such that $\alpha + \beta$, and $\alpha\beta$ are algebraic. Show that $\alpha$ and $\beta$ are algebraic. attempt: Suppose $\alpha, \beta \in \mathbb{C}$ such ...
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### What is this notation? $V(\Bbb Z/p\Bbb Z)$

I'm trying to write a blog post, and I've run into a stumbling block with notation. Is $V(\Bbb Z/p\Bbb Z)$ a standard notation in algebraic number theory? Does it mean a variety restricted to the ...
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### Can there be a set of numbers, which have properties like those of quaternions, but of dimension 3? [duplicate]

We all know what the complex numbers are : basically $$R^2$$ with a specified product formula, which is $$(a,b).(c,d)=(ac-bd,ad+bc).$$Hamilton defined the quaternions, which are a wonderful set of ...
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### Ideals in a Dedekind domain localized at a prime ideal

Let $R$ be a Dedekind domain, let $\mathfrak{i}$ be a non-zero ideal of $R$. By factorization theorem we can write $$\mathfrak{i}=\mathfrak{p}_1^{a_1}\cdots\mathfrak{p}_n^{a_n}$$ for distinct non-zero ...
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### Confusion about proof of necessary + sufficient condition for $\theta \in \mathbb{C}$ to be an algebraic integer

I am confused about a step in a proof of the following statement in Stewart Tall's ANT book: $\theta \in \mathbb{C}$ is an algebraic integer if and only if the additive group $G$ generated by powers ...
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### What is the minimal polynomial of $\bigotimes_{j=0}^{\infty}\mathbb{Q}(\zeta_{j})$? [closed]

What is the minimal polynomial over $\mathbb{Q}$ of $\bigotimes_{j=0}^{\infty}\mathbb{Q}(\zeta_{j})$, where $\zeta_j$ is a $j$-th primitive root of unity for each $j$? I want to say it should be ...
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### Abelian extension of an algebraic number field whose Galois group is isomorphic to a given finite abelian group

Let $K$ be an algebraic number field, i.e. a finite extension of $\mathbb{Q}$. Let $G$ be a finite abelian group. Does there exist a Galois extension of $K$ whose Galois group is isomorphic to $G$? I ...
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### No ideals coprime in DVR?

There's this exercise in Neukirch, chapter I, §3 (i've changed the statement to deal only with the case that bothers me): Let $\mathfrak o$ be a Dedekind domain and $\mathfrak m$ be a nonzero ...
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### ideal calculation and relations

Let $f$ be an integral ideal of a number field $K$ (with ring of integers $\mathcal{O}$ and let $a$ and $b$ be fractional ideals of the same. Suppose that $ab^{-1} = x\mathcal{O}$ for some $x \in K$ ...
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### Problem with units in number field

Edit:There were several major mistakes by my side this post, most of which have been accounted for.Now, after editing these out, the post seems to have no purpose at all.Nevertheless, it feels wrong ...
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### Relation between a quadratic residue and it's order

In the context of the multiplicative group $(\mathbb{Z}/m\mathbb{Z})^\times$ of congruence classes modulo $m$ coprime with $m$, is there a theorem that states something about the order of a given ...
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### Variant of strong approximation.

Let $K$ be a global field. Let $w$ be a place of $K$. Let $\textbf{A}^w$ be the restricted direct product over all $v$ except $w$ of the $K_v$ with respect to the subgroups $\mathcal{O}_v$. How do I ...
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### Give a basis for the ring of integers for K, a finite extension of Q in C

This question concerns the diophantine equation $$x^2+87y^2=47z^2$$ we set $K=\mathbb{Q}(\sqrt{-89})$. Define the ring of integers $o_{K}$ of K and give a $\mathbb{Z}$-basis of the form {$1,\alpha$} ...
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### Is $\widehat{K}L$ complete?

Let $K$ be a field and $\widehat{K}$ be a completion with respect to some valuation on $K$. Let $L$ be a finite separable extension of $K$. When regarded as a subfield of $\widehat{L}$, is ...
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### Is $\mathbb{A}_Q/\mathbb{Z}$ compact and connected?

Let $\mathbb{A}_\mathbb{Q}$be the the adéles of $\mathbb{Q}$ viewed as a group and consider $\mathbb{Z}$ as a subgroup in the natural manner. Is $\mathbb{A}_\mathbb{Q} / \mathbb{Z}$ compact and ...
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### Number field extension

Given a number field K show that there exists a number field extension L of K such that every ideal in K becomes a principal ideal in L.
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### Two algebraic number theory questions.

I have two algebraic number theory questions. See page 26 here, specifically Lemma 5.11. Lemma 5.11. The group $\mathcal{O}^\times$ is generated by roots of unity and $[\mathcal{O}^\times]^+$, the ...
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### Proving that a Prime Ideal Divides a Principal Ideal

Let $K/\mathbb Q$ be a cubic extension and $(p) = \mathfrak p_1 \mathfrak p_2 \mathfrak p_3$ be a factorization into distinct prime ideals. Suppose that $\alpha \in \mathcal O_K$ is an integer such ...
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### On orders of a quadratic number field

Let $K$ be an algebraic number field of degree $n$. An order of $K$ is a subring $R$ of $K$ such that $R$ is a free $\mathbb{Z}$-module of rank $n$. Let $\alpha_1, \cdots, \alpha_n$ be a basis of $R$ ...
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### Topology for the decomposition of $L\otimes_K K_\frak p$

Let $L/K$ be an extension of number fields, $\frak P|\frak p$ primes of $\mathcal O_L$ and $\mathcal O_K$, and let $K_\frak p$ be the completion of $K$ wrt $|\cdot|_\frak p$ and $L_\frak P$ the ...
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### Prime divisors of the conductor of an order of an algebraic number field

Let $f(X) \in \mathbb{Z}[X]$ be a monic irreducible polynomial. Let $\theta$ be a root of $f(X)$. Let $A = \mathbb{Z}[\theta]$. Let $K$ be the field of fractions of $A$. Let $B$ be the ring of ...
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### How to prove a generalized Gauss sum formula

I read the wikipedia article on quadratic Gauss sum. link First let me write a definition of a generalized Gauss sum. Let $G(a, c)= \sum_{n=0}^{c-1}\exp (\frac{an^2}{c})$, where $a$ and $c$ are ...
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### Different discriminant ideal, what are their applications?

Lots of texts online on number theory do not even mention the different ideal. Some do, but then it gets ignored and is never mentioned again. I could not find a single application for it, as if it is ...
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### Proving unit of quartic number field is fundamental

Let $K = \mathbb Q(\alpha)$, for $\alpha$ a root of $a^4 + 4 \alpha^2 + 2 = 0$. I want to prove the group of units $\mathcal O_K^*$ equals $\langle -1, \alpha^2 + 1\rangle$. I've found the ring of ...
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### polynomial with all roots on the unit circle

I'm wondering if the following statement is true: if all roots of a polynomial with integer coefficients are on the unit circle, then these roots are in fact roots of unity and the polynomial is a ...
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### kernel of the artin map when dealing with S-ideles and S-divisors for function fields

Let $L/K$ be an abelian extension of function fields and let $\vartheta_{L/K}:\mathcal{D} \to \text{Gal}(L/K)$ be the Artin map from the divisors of $K$ to the galois group. What can be said about ...
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### Average of sum of unit roots is an algebraic integer

Let $\alpha_1,\ldots, \alpha_n$ be roots of unity, and let $a=\frac{1}{n}\sum\alpha_i$. Then if $a$ is an algebraic integer, we have either $a=0$ or $a=\alpha_1=\dots=\alpha_n$. Why?
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### In what way is a quadratic extention to a finite field isomorphic to a finite field of higher order?

I have read (I don't remember where) that a finite field that is quadratically extended, say $\mathbb F_p[\sqrt 3]$ for example, is isomorphic to the finite field $\mathbb F_{p^2}$ (assuming the ...
$\xi = \cos{\frac{ 2\pi}{n}}+i \sin{\frac{ 2\pi}{n}}$ , $i^2=-1, n$ is a positive integer. if $\xi^{a_1}+\xi^{a_2}+...+\xi^{a_k}=0$ , $a_1,a_2,...,a_k\in \{0,1,...,n-1\}$ and $a_1,a_2,...,a_k$ are ...