Questions related to the algebraic structure of algebraic integers

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Bounding height of an algebraic number

Let $\zeta_l$ be a primitive $l^{th}$ root of unity($l\geq 5$) and let $H$ be the height of an algebraic number(see for instance page 230 of The Arithmetic of Elliptic Curves by J. Silverman). I ...
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References for Algebraic number theory

I am doing algebraic number theory first time. I have done all ring theory and field theory. I am interested in algebra , so also pretty much excited about algebraic number theory. I have a month's ...
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Specific question on dirichlet density

In a notes I found the following exercise and solution: I have a question. In the proof I admit the statement "the Dirichlet density of these prime ideals is $1/2$ " but i do not understand why the ...
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Representation groups over Dedekind domains

I am interested on groups defined over $O_K$ the ring of integers of a number field $K$. Given a linear representation $T:Gl_N(O_K)\rightarrow Gl(W)$ with $W$ a free $O_K$-module, What are the main ...
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Isomorphism for the group of units of the ring of integers of a local field

Let $K$ be a local field with a discrete and non-archimedean absolute value, $\mathcal{O}_K$ be its ring of integers, $\mathfrak{m}_K$ be the unique maximal ideal of $\mathcal{O}_K$ and ...
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Galois theory on curves

Context: Let $\mathbb{F}$ be the algebraic closure of $\mathbb{F}_q$ for $q$ prime. We know that $\mathbb{F}(t)$ for $t$ transcendental is the function field of the projective line ...
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Given that there is at least one solution to $a^{2} + 2b^{2} = p^{11}q^{13}$, find how many integers solutions there are.

I cannot even begin this problem, given $ a, b \in \mathbb{Z}$ and $p,q$ odd prime numbers, given that there is a soltuion to the equation: $a^{2} + 2b^{2} = p^{11}q^{13}$, find how many solutions ...
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Why does taking completions make number fields simpler?

I'm currently taking a course on Local Fields, and the local-theoretic picture seems to be significantly simpler than that of number fields. For example, If $K$ is a finite extension of $\mathbb ...
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Special case of Kronecker–Weber theorem.

Let $K$ be a number field contained in $m^{th}$ cyclotomic field, that is $K \subset \Bbb{Q}(\omega)$ where $\omega$ is a primitive $m^{th}$ root of unity. Let $p^k$ be the exact power of a prime $p$ ...
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1answer
27 views

Nontrivial characters of $(\Bbb{Z}_m)^{\ast}$

I was reading the book of Marcus on Number field page 196. I could not understang the highlighted equality. It will be helpful if someone gives me a proof. Thanks in advance for the computation!
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How to calculate the ideal class group of a quadratic number field?

The books I use to study Algebraic Number Theory are rather thin on the ground with concrete examples, so I make my own and check the results with Sage. To get some more hands on experience I want ...
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Primes Representable as Quadratic Form - Specifically Norm of an Imaginary Quadratic Field with Class # 1

Could someone point me towards the result that states that a prime is expressible as a norm of an imaginary quadratic field with class number 1 iff $\left(\frac{p}{D}\right)=1$.
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Is the ring of p-adic integers of finite type over the ring of integers?

Denote by $\mathbb{Z}_p$ the ring of $p$-adic integers. Is $\mathrm{Spec}(\mathbb{Z}_p)$ of finite type over $\mathrm{Spec}(\mathbb{Z})$?
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another representation of the zeta function of a curve over a finite field

Let $C$ be a non-singular curve over $\mathbb{F}_q$. Denote by $d$ the degree map from the group of divisors to $\mathbb{Z}$ and denote by $P$ the set of prime divisors w.r.t. to the function field. ...
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coefficients of the zeta function of curve over a finite field $\mathbb{F}_q$

Let $C$ be a non-singular curve over $\mathbb{F}_q$. Denote by $d$ the degree map from the group of divisors to $\mathbb{Z}$ and denote by $P$ the set of prime divisors w.r.t. to the function field. ...
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Is it true that $B=[\beta]$ when $B$ is and ideal of $\mathcal{O}$, $\beta \in B$ and $N(B)=|N(\beta)|$?

Let $B$ be an ideal of $\mathcal{O}$ (Ring of integers), $\beta \in B$ and $N(B)=|N(\beta)|$. Does it follow that $B=[\beta]$? I think that this isn't true but I'm struggling to find a counter ...
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187 views

Ring of integers is not a field

Let $K$ be a finite extension of $\mathbb{Q}$ (i.e. $K$ is a number field). Let $\mathbb{B}$ be the set of all algebraic integers. Inside $K$, we have the so-called the ring of integers ...
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An easy question about fractional ideals…

Let $A$ be an integral domain and $K$ its field of fractions. If $M$ is a non-zero fractional ideal of $A$, then $$N=\{x \in K : xM \subseteq A\}$$ is also a fractional ideal of $A$. The proof I am ...
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How can you write $\mathbb{Q}[\sqrt[3]{2},\omega_3]$ using a single algebraic element $\mathbb{Q}[\alpha]$?

Looking at the basis of $\mathbb{Q}[\sqrt[3]{2},\omega_3]$ gives me no idea on how to generate it using $\{1, \alpha, \alpha^2,\alpha^3,\alpha^4,\alpha^5\}$ for some $\alpha$ algebraic over ...
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How to factor the ideal $(65537)$ in $\mathbb Z[i]$?

This question is related to How to factor ideals in a quadratic number field? In Algebraic Number Theory by W. Stein he makes a remark about the factorization of $65537$ in $\mathbb Z[i]$. I ...
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Counting tamely ramified Galois extensions of $\mathbb{Q}_p$ with a given Galois group.

For a homework exercise, I'm to determine for each $p$ the number of non-isomorphic tamely ramified Galois extensions $K/\mathbb{Q}_p$ such that $\operatorname{Gal}(K/\mathbb{Q}_p) \cong ...
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1answer
69 views

How to factor ideals in a quadratic number field?

Let $w=\sqrt{-5}$; we work in $\mathbb Z[w]$; which is not a PID or UFD. Now in Sage (3) factors in $(3,1+w)(3,2+w)$ and (7) in $(7,3+w)(7,4+w)$. Please explain clearly how factorization of ideals ...
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Motivation and examples for ramification

I started learning algebraic number theory, but it seems like all the sources I had are too abstract, giving me difficulty understanding the concept and tripping me up frequently. For today it is ...
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108 views

Why do we have to work to prove the surjectivity of the local Artin map (Serge Lang A.N.T., Chapter XI Theorem 3)

I must be misunderstanding something about Artin reciprocity. Let $K/k$ be an abelian extension of number fields with Galois group $G$, $I_k$ the ideles of $k$, and $P$ a prime of $k$ (with $v$ a ...
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Fractional ideals of $\mathbb{Q}$ prime to $N$

Let $N \in \mathbb{Z}$. What is meant by a fractional ideal $\mathfrak{p}$ of $\mathbb{Q}$ being prime to $N$? Is it that $gcd(\mathfrak{p},N\mathbb{Z})$ contains $\mathbb{Z}$? Let $I_N$ denote the ...
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What is a concrete example to demonstrate that $\mathcal{O}_{\mathbb{Q}(\sqrt{-19})}$ is NOT a norm-Euclidean domain?

I'm talking about the ring of algebraic integers of the form $\frac{a}{2} + \frac{b \sqrt{-19}}{2}$, where $a, b \in \mathbb{Z}$. This ring is said to be a principal ideal domain but not a Euclidean ...
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1answer
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Nice proof that $\mathbb{Z}[\sqrt{6}]$ is a Euclidean domain wrt absolute norm map

I know that $\mathbb{Z}\left[\sqrt{6}\,\right]$ is a Euclidean domain with respect to the absolute valued norm map $x+y\sqrt{6} \mapsto |x^2-6y^2|$. I think I proved this result with some common ...
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What is the 'Hom-description'?

I am trying to learn about the 'Hom-description' of the class group $Cl(A)$ of an $R_K$-order $A$ in $K[G]$ where $K$ is a number field with ring of integers $R_K$ and $G$ is a finite group. I've ...
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Eisenstein integers and applications to Diophantine equations

Solve the equation $7\times 13\times 19=a^2-ab+b^2$ for integers $a>b>0$. How many are there such solutions $(a,b)$? I know that $a^2-ab+b^2$ is the norm of the Eisentein integer $z=a+b\omega$, ...
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1answer
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Lattice with conditions

In Book's Algebraic number theory, I. Stewart page 142: Theorem 7.2: If $p$ is prime of the form 4k+1 then $p$ is sum of two squares. Proof: The multiplicative group $G$ of the field $\mathbb ...
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Ramification index multipicative

Let $R\subseteq R'\subseteq R''$ be Dedekind rings and P a non-zero prime ideal in $R''$ .I need to show that $e(P/R)=e(P/R')e(P\cap R'/R)$ where $e(P/R)$ is the ramification index of P in respect of ...
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Can we tell if a number is prime by the number of its partition ?

Can we tell if a number is prime by the number of its partition ? Or in general, how much can we know about a number itself from its partition function ? I understand that Ramanujan has some ...
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Quotient ring of the ring of integers of an algebraic number field and its fraction field

Let $K$ be an arbitrary algebraic number field. We know that the fraction field of $\mathcal{O}_K$ is $K$ which is always isomorphic to some $\mathbb{Z}[x]/(f(x))$. $\mathcal{O}_K$ also has dimension ...
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How to show that if $\gamma \alpha=0$ (mod $A$) then $\alpha = 0$ (mod $A$)

Let $A$ be an ideal of $\mathcal O$ (Ring of integers of some algeibraic number field) and assume that $gcd([\gamma],A)= [1]$. How to show that if $\gamma \alpha=0$ (mod $A$) then $\alpha = 0$ (mod ...
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Solve $3^a-5^b=2$ for integers a and b.

So I have got that (a,b)=(1,0),(3,2) are solutions for the eqations, and maybe the only one.
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1answer
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Typo in the book or am I going crazy?

I am reading about integral bases from Frazer Jarvis' "Algebraic Number Theory", but my question is really about elementary linear algebra. In page 49, author claims the following: I don't think ...
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Prime ideals of the ring of integers of an algebraic number field

I am working on a problem that has a completely different point and I didn't work with algebraic number fields much before, so I was wondering if someone could point me in the right direction for ...
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can anyone give a proof by definition: $11$ is prime in $ \mathbb{Z}[\sqrt{-5}] $

what i did is: assume $\alpha \notin (11),\beta\notin (11), \alpha\beta \in (11)\Rightarrow\exists \gamma, s.t.$ $ \alpha\beta = 11 \gamma$, $\Rightarrow N(\alpha)N(\beta) = 11^2N(\gamma) $ then ...
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Norm in algebraic number fields

Consider an algebraic number field $\mathbb{Q}(\alpha)$ and it's ring of integers $O$. If we take any element $\xi \in O$ and we want to calculate it's norm $N_{\mathbb{Q}(\alpha)/\mathbb{Q}}(\xi)$, ...
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ramification index of $p$ in $\mathbb{Z}\left[e^{\frac{2\pi i}{p}}\right]$

I am attempting to show that $p$ has ramification index $p-1$ in $\mathbb{Z}[\omega]$ where $\omega=e^{2\pi i/p}$. The issue is I want to do so avoiding actually factoring $p$. I was hoping to use ...
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Help unmasking a disguised principal ideal

I recently saw a question on here about trying to generate a non-principal ideal in a principal ideal domain, with the only answer so far saying that if the ring $R$ is a PID, then $\langle e, f ...
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Algebraic number field with non trivial integral basis

So far I have only seen extensions of $\mathbb{Q}$ with "trivial" integral basis. Meaning that the integral basis is the most natural one e.g. the integral basis for $\mathbb{Q}(\sqrt[3]{2})$ is just ...
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Show that there are two ideal classes in $\mathbb{Z}[\sqrt{10}]$

Show that there are two ideal classes in $\mathbb{Z}[\sqrt{10}]$. I'm trying this problem with the Minkowski bound, please I need more help. Thanks
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In a PID, does every attempt to generate a non-principal ideal just lead back to the whole ring itself?

It is a well-known fact that a unique factorization domain is a principal ideal domain, in which all ideals are principal ideals. [EDIT: I got dyslexic on this one, should've said something along the ...
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1answer
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Field, algebraic element

1) Let $E/F$ an extension and let $\alpha,\beta\in E$ be algebraic elements over $F$. If $\alpha\neq 0$, prove that $\alpha+\beta$, $\alpha\beta$ and $\alpha^{-1}$ are all algebraic over $F$. 2) If ...
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Show that $x$, $y$, $z$ are integers when $3x$, $3x^2-6yz$, $x^3+2y^3+4z^3-6xyz$ are integers.

I was trying to show that $\{1, \alpha, \alpha^2\}$ is a integral basis of $\mathbb{Q}(\alpha)$ where $\alpha= \sqrt[3]{2}$. And after some steps it remains to prove that if $$3x, \quad 3x^2-6yz, ...
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Does the equation $\tan(x)=y$ have any non-zero rational solution?

Trivially $\tan(0)=0$ but it seems this is the "unique" solution of the equation $\tan(x)=y$ on rational numbers. In fact if we try to make $y$ rational we usually use irrational (transcendental) ...
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Lattices in $\mathbb C$ as modules of the ring of integers in an imaginary quadratic field

Let $K$ be an imaginary quadratic number field and let $O_K\subset K$ be the ring of algebraic integers in $K$. Let us call a lattice $\Lambda\subset\mathbb C$ normalized if the tori $\mathbb ...
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What is the class number of $\mathcal{O}_{\sqrt[3]{18}}$?

I accept it without proof that $\mathcal{O}_{\sqrt[3]{2}}$ and $\mathcal{O}_{\sqrt[3]{3}}$ both have class number $1$. Also, I've been told that $\mathcal{O}_{\sqrt[3]{m^2}} = ...
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1answer
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$\sum_{\zeta^p=1}(\zeta-1)^n$

Given $n\geq0$ let $$ z_n=\sum_{\zeta^p=1}(\zeta-1)^n $$ where $p$ is an odd prime number (summation extended to all $p$-th roots of 1). It is clear that: $z_n\in\Bbb Z$ (it's a Galois invariant sum ...