Questions related to the algebraic structure of algebraic integers

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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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Nice Proof $\mathbb{Z}[\sqrt{6}]$ is a euclidian domain wrt absolute norm map

I know that $\mathbb{Z}\left[\sqrt{6}\,\right]$ is a Euclidian 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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3answers
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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
19 views

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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22 views

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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1answer
32 views

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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23 views

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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1answer
25 views

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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1answer
118 views

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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0answers
102 views

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 ...
2
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1answer
43 views

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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1answer
122 views

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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2answers
34 views

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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1answer
43 views

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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1answer
28 views

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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1answer
41 views

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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1answer
55 views

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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1answer
38 views

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
30 views

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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1answer
47 views

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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1answer
47 views

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
80 views

$\sum_{\zeta^p=1}(\zeta-1)^n$

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 ...
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1answer
29 views

Completion of a number field at a complex embedding

Sorry if this question has been asked before. Let $K$ be a number field of degree $n>1$ and $\sigma:K\hookrightarrow \mathbb C$ a complex (non real) embedding of $K$ in $\mathbb C$ giving the ...
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0answers
28 views

Identity relating Frobenius to multiplication by p on the ring of Witt vectors

Let $k$ be a field of characteristic $p$, $F$ be the Frobenius map of Witt vectors, and $V$ the transfer map on $W(k)$. I'm trying to show that $FV(a) = pa$ where $a$ is a Witt vector. Clearly this is ...
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4answers
92 views

Uniqueness of representation of prime as $x^2+2y^2$

It can be proven that every prime $p\equiv1,3\mod{8}$ can be written in the form $a^2+2b^2$. Is it true that this representation is unique? This is certainly true for primes written in the form ...
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1answer
28 views

Density of ring of algebraic integers in $\mathbb C$

Clearly, $\mathbb Z$ is not dense in $\mathbb Q$ (and not dense in $\mathbb C$). But why is the ring of algebraic integers $\overline{\mathbb Z}$ dense in $\overline{\mathbb Q}$? In particular, if ...
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Exercise 2.8 Cassels and Frohlich

I don't understand the discussion in exercise 2.8 of Cassels and Frohlich (page 352) beginning with "more generally". Why should it matter whether the formula for $c$ has a power of $-1$ in it if this ...
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0answers
30 views

Class number 1 for negative integers

If n is a positive integer then, $h(-4n) = 1 \iff n = 1,2,3,4,7$. The only proof I have seen of this is long and case wise. Is there any conceptual a priori reason to believe that these numbers are ...
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1answer
40 views

How can I see that $4$ is not a quartic residue?

How can I see that $4$ is no quartic residue, i.e. there is no $t$ such that $t^4 \equiv 4 \mod p$ when $p\equiv 5 \mod 8$?
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1answer
49 views

Norm of Prime Ideal

Show that the norm of a prime ideal in a number field $K$ is a power of some prime number, i.e., if $P$ is a prime ideal in $O_K$ for some number field $K$, then $N_\mathbb{Q}^K(P)=p^n$ for some ...
3
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1answer
23 views

Extending a DVR could produce not a DVR

I'm reading Tate's paper about $p$-divisible groups. In Chapter $(2.4)$ he asserts that if you take $R$ a complete DVR with residue field $k$ of characteristic $p>0$, $K$ its field of fractions, ...
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Irreducibility of cyclotomic polynomials via schemes

A few months ago, someone told me there existed a scheme theoretic proof of the irreducibility of cyclotomic polynomials. I've tried coming up with a proof, and when that didn't really yield anything ...
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1answer
22 views

Is it true that an equivalent 'absolute value' is an absolute value?

I've a very basic question on absolute values on fields. If $K$ is a valued field with absolute value $|- |:K\to \mathbb R_{\geq0}$ then is the map $|-|':K\to \mathbb R_{\geq0}$ defined by ...
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Show that $\sqrt [3]{2}-\sqrt [3]{4}$ is algebraic

How do I show, step by step, that $\sqrt [3]{2}-\sqrt [3]{4}$ is a root of $x^3+6x+2$? Start with $x=\sqrt [3]{2}-\sqrt [3]{4}$ do not use the cubic, the cubic is given for convenience. ( This is ...
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1answer
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the number of non-zero integral ideals of norm m in a ring of integers [closed]

How to prove that the number of non-zero integral ideals of norm m in a ring of integers of a number field with degree n is less than or equal to the number of n-dim vectors of n positive integer ...
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61 views

An Efficient Route to Tate's Thesis

I want to learn Tate's thesis. My advisor suggested the Book "Algebraic Number Theory" By Lang. However, it seems to be a long read before I reach Tate's Thesis. I want to know what are other good ...
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Finding the Extension Degree of a Cyclotomic Field

Greetings Mathematics Community. I am having much difficulty in solving the following problem: If $m\equiv 2$ (mod 4), show that $\mathbb{Q(\zeta_m)}=\mathbb{Q(\zeta_{\frac{m}{2}})}$ where $\zeta$ ...
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1answer
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Find the Value of $n$ Where $15756$ is the $nth$ Member of A Set

It's a question from $BNMO$.It still haunts me a lot. I want to find an answer to this question. Any number of the different powers of $5: 1,5,25,125$ etc is added one at a time to generate the ...
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0answers
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Is the group $I_K/K^{\ast}$ compact?

I have two question on adeles and ideles: $1)$Let $K$ be a number field. Is the group $I_K/K^{\ast}$ compact? Here $I_K$ is the idele group of $K$. $2)$ Also it will be helpful if someone explains ...
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Connected component of the Idele group

Let $K$ ba a number field with $r_1$ real embeddings and $r_2$ pairs of complex embeddings. Let $I_K$ be the group of ideles of $K$ and let $H$ be the connected component of identity. How to show that ...
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1answer
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valuation ring, completeness

Perhaps a trivial question: is there an example of a field $K$ and a valuation $v$ on $K$ such that the following holds: $K$ is not complete (with respect to the valuation topology) The valuation ...
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1answer
99 views

$x^{16}-16 \equiv 0 \mod p$ has a solution for each prime

I have to prove that $x^{16}-16\equiv 0 \mod p$ has a solution for every prime $p$. I already know (from a previous work) that $x^8-16\equiv 0 \mod p$ has a solution for every prime. In my opinion, I ...
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1answer
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sequence $\{a^{p^{n}}\}$ converges in the p-adic numbers.

Let $a\in \mathbb{Z}$ be relatively prime to $p$ prime. Then show that the seqeunce $\{a^{p^{n}}\}$ converges in the $p$-adic numbers. This to me seems very counter intuitive. Since $(a,p)=1$ the ...
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Topology on ideles

I am new to adeles and ideles in number fields. I am trying to solve Exercise 2 in this pdf. This is a very standard fact the statement of which can be found in any textbook. I have done exercise 2 ...
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1answer
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Confusion on Inert Primes in Ireland and Rosen

In Ireland and Rosen, the following law for inert rational primes in a quadratic field is stated as: if $p\nmid \delta_K$, where $\delta_K$ is the discriminant of the quadratic field, and $d$ is a ...
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1answer
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Kummer-Dedekind's factorisation theorem

For a number field extension $K$ of $\mathbb{Q}$ one can factorise almost all prime ideals $(p)$ in the extension $K$, except finitely many, easily by factorising minimal polynomials in finite ...