Questions tagged [integer-rings]
In algebraic number theory, the ring of integers of a number field $K$ is the ring of all elements of $K$ which are roots of a monic polynomial with rational integer coefficients.
6
votes
1answer
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Ring of integers of $\mathbb{Q}(i,\sqrt{5})$
I'm trying to find the ring of integers $A_L$ of $\mathbb{Q}(i,\sqrt{5})$. I know that the ring of integers of $\mathbb{Q}(i)$ is $\mathbb{Z}[i]$ and that the one of $\mathbb{Q}(\sqrt{5})$ is $\mathbb{...
3
votes
1answer
61 views
When will a prime element of $\Bbb{Z}[(\sqrt{5}-1)/2]$ have field norm equal to a rational prime?
Consider the integer ring of $\mathbb{Q}[\sqrt{5}]$, i.e. $\mathbb{Z}[(\sqrt{5}-1)/2]$. Then if $N(x)$ denotes the field norm of $x\in\mathbb{Z}[(\sqrt{5}-1)/2]$, then $N(x) = p$ for a rational prime $...
1
vote
1answer
58 views
$n\in \mathbf{N}$ such that a solution of $X^4+nX^2 +1$ is a root of unit
Consider $f_n(X)=X^4+nX^2 +1$ in $\mathbf{Q}[X]$. I found that for all natural $n$ such that $n\neq 2-m^2$ for a natural $m$, $f_n(X)$ is irreducible in $\mathbf{Q}$.
Consider $K_n=\mathbf{Q}(x)= \...
2
votes
1answer
67 views
the ring of integers: prove that (2) is a prime ideal and that it is a pid
Consider a real root $\alpha$ of $f(X)=X^3-3X+1$. Consider the ring of integers $A_K$ for $K=\mathbf{Q}[\alpha]$. I showed that the ideal $(1+\alpha)$ is prime in $A_K$ and that $A_K=\mathbf{Z}[\alpha]...
2
votes
1answer
72 views
Clarification about the definition of ring of integers of a local field
Classically, the ring of integers of a number field $K / \mathbb{Q}$ is defined to be the collection (forms a ring) of those elements $\alpha \in K$ such that there is a monic $f \in \mathbb{Z}[x]$ ...
0
votes
0answers
61 views
Basis of the integer ring of $\mathbb{Q}(\sqrt5)$
I am solving an exercise with the field extension $\mathbb{Q}(\sqrt5)/\mathbb{Q}$
I am suck trying to prove that:$$B = \{a+b(\dfrac{1+\sqrt5}{2}) |a,b\in \mathbb{Z} \}$$
Where B is the integer ring ...
2
votes
2answers
35 views
Evaluate an expression as an element of $F$
Let $f(x)=2x^{2}+2x+1\in \mathbb{Z}_{3}[x]$
The first part of the problem i'm trying to solve was to prove that $$\frac{\mathbb{Z}_{3}[x]}{f(x)\mathbb{Z}_{3}[x]}$$ is a field. (stating clearly any ...
0
votes
1answer
39 views
Is the integral closure of a ring of integers in finite separable extension a ring of integers?
Let $K/F$ be a finite separable extension of number fields of finite degree over $\mathbb{Q}$. Let $A = \mathcal{O}_F$ and $B$ the integral closure of $A$ in $K$. Is $B = \mathcal{O}_K$?
Let us ...
1
vote
2answers
34 views
Characterizing elements with square norms in quadratic integer rings
Given the ring $\mathbb{Z}\left[\sqrt{D}\right]$ (where $D$ is a positive square-free integer) can we characterize all elements $\alpha$ with positive norms for which $N(\alpha)$ is a perfect square ...
0
votes
1answer
70 views
List all ideals of integer ring of norm less or equal 10 [closed]
I have the following exercise:
The cubic field with the smallest discriminant, in absolute value, is $\mathbb{Q}(\alpha)$ with $\alpha$ a root of $T^3-T+1$ and with ring of integers $\mathbb{Z}[\...
1
vote
1answer
49 views
Is the indicated subring an order?
I have an exercise in my class that I don't seem to find an answer to:
We are asked to tell whether the indicated subring of a number field is an order and whether it is a maximal order
a) $\mathbb{...
8
votes
5answers
120 views
Is $1+\sqrt{5}$ a prime under the $\mathbb{Z}[{\sqrt{5}}]$ domain?
The title is self-explanatory. I know it's irreducible but is it a prime? How to prove these primality and/or irreducibility of $1+\sqrt{5}$.
Can you just briefly state how a prime is defined under $\...
2
votes
1answer
77 views
Describing number ring corresponding to $\mathbb{Q}(\sqrt{p_1},…,\sqrt{p_k})$
Let $p_i$'s be distinct primes such that $p_i \equiv 1\;(\mathrm{mod}\; 4)$ for every $i=1,...,k$. It is well-known that $\dfrac{1+\sqrt{p_i}}{2}$'s are algebraic integers and the number ring $\mathbb{...
2
votes
0answers
60 views
$\operatorname{disc}(\mathbb{Z}[\alpha]) = [\mathcal{O}_K:\mathbb{Z}[\alpha]]^2\operatorname{disc}(\mathcal{O}_K)$.
Let $\alpha\in\mathcal{O}_K$ such that $K=\mathbb{Q}[\alpha]$. Define $\operatorname{disc}(\mathbb{Z}[\alpha]) := \operatorname{disc}(1,\alpha,\dots,\alpha^{n-1})$. Show $\operatorname{disc}(\mathbb{Z}...
6
votes
3answers
97 views
Why is $\mathcal{O}_K$ the ring to be considered for factorizing integers?
For $K = \mathbb{Q}[\alpha]$ (with $\alpha$ algebraic over $\mathbb{Q}$), I understand that $\mathbb{Z}[\alpha]$ may be too coarse, and that $\mathcal{O}_K$ (the algebraic integers of $K$) allows more ...
9
votes
4answers
848 views
Is a ring of integers necessarily Noetherian?
Let $K$ be an algebraic extension field (not necessarily finite) of $\mathbb{Q}$. Let $\mathscr{O}_K$ be the integral closure of $\mathbb{Z}$ in $K$.
Then, is $\mathscr{O}_K$ Noetherian?
If $K$ is a ...
12
votes
1answer
519 views
When is the tensor product of rings of integers again a ring of integers?
Given number fields $K$ and $L$, under what conditions does there exist a number field $M$ such that
$$\mathcal{O}_K\otimes_{\Bbb{Z}}\mathcal{O}_L\cong\mathcal{O}_M.$$
It is necessary that $K$ and $L$ ...
0
votes
1answer
32 views
Product of all embbedings of a fractional ideal is principal
Let $F$ be a number field of degree $n$ and $\mathfrak a \subset F$ a fractional ideal. How do you show that
$$\prod_{k=1}^n \sigma_k(\mathfrak a)$$
is in the trivial ideal class with a rational ...
1
vote
1answer
42 views
Why $\mathbb{Z}[\theta]\,/\,\mathcal{P}$ is an algebraic extension over $\mathbb{Z}/p\mathbb{Z}$?
If $f(x)$ is a monic, irreducible polynomial in $\mathbb{Z}[x]$ with $\theta\in\mathbb{C}$ as root, why $\mathbb{Z}[\theta]\,/\,\mathcal{P}$ is an algebraic extension over $\mathbb{Z}/p\mathbb{Z}$?
I'...
2
votes
2answers
55 views
Why $\mathbb{Z}[\theta]\,/\,\mathcal{P} \simeq \mathbb{F}_{p^e}$ for any non-zero prime ideal $\mathcal{P}$ of $\mathbb{Z}[\theta]$?
If $f(x)$ is a monic,irreducible polynomial in $\mathbb{Z}[x]$ with $\theta\in\mathbb{C}$ as root, why $\mathbb{Z}[\theta]\,/\,\mathcal{P} \simeq \mathbb{F}_{p^e}$ for any non-zero prime ideal $\...
1
vote
1answer
226 views
Existence of a fundamental solution to the Pell's equation
Let $\mathbb{Z}[\sqrt{3}]$ be the ring $\{a + b\sqrt{3} | a, b \in \mathbb{Z}\}$ with a natural structure. We consider it's norm:
$$a + b\sqrt{3} \mapsto a^2 - 3b^2$$
Then, finding the solution to ...
3
votes
1answer
160 views
Show that algebraic integers in $\mathbb{Q}[\zeta_3]$ are exactly $\mathbb{Z}[\zeta_3]$.
If $\zeta_3$ is a primitive cube root of unity, then I'm trying to show that $\mathbb{Z}[\zeta_3]$ is the ring of algebraic integers in $\mathbb{Q}[\zeta_3]$. I have shown that $\mathbb{Z}[\zeta_3]$ ...
2
votes
1answer
42 views
Question regarding fractions of algebraic integers
Suppose that $K$ is an extension field of $\mathbb{Q}$ and that $\alpha \in K$ is a non-zero algebraic integer (i.e. it is integral over $\mathbb{Z}$). I am trying to show that there are only finitely ...
5
votes
0answers
81 views
Calculating discriminant of elements of $\mathbb{Q}(a)$
Could someone please guide me through this one?
Let $a$ be a root of the irreducible polynomial $x^3+rx+s \in \mathbb{Z}[x]$.
Calculate: $disc(1,a,a^2)$.
Thanks in advance
1
vote
0answers
113 views
Structure of Ring of Integers
Every finite extension $K$ of $\mathbb Q$ can be written as $\mathbb Q[\alpha]$. A very naive but important question is to ask if the ring of integers of $K$ is equal to $\mathbb Z[\alpha]$. When is ...
3
votes
1answer
144 views
Primes, integers and fundamental units in cubic fields.
Self taught here so please bear with me. How does one define the ring of integers of the field $\mathbb{Q}(r)$, where $r$ is a root of the cubic $$x^3+px+q$$ as well as determining the fundamental ...
10
votes
2answers
360 views
Finding the ring of integers of $\Bbb Q(\sqrt[4]{2})$
I know$^{(1)}$ that the ring of integers of $K=\Bbb Q(\sqrt[4]{2})$ is $\Bbb Z[\sqrt[4]{2}]$ and I would like to prove it.
A related question is this one, but it doesn't answer mine.
I computed ...
4
votes
1answer
171 views
Determining when ring of integers is $\mathbb{Z}[\theta]$
Something which is not difficult to prove is that if $K$ is a number field generated by an integer $\theta$, then the ring of integers $\mathfrak{O}_K$ is generated over $\mathbb{Z}$ by $\theta$ and ...
1
vote
4answers
75 views
$p$ is a positive integer and $(p)$ is a maximal ideal in the ring $(\mathbb Z, +,\cdot)$, then $p$ is a prime number
I need to prove: $p$ is a positive integer and $(p)$ is a maximal ideal in the ring $(\mathbb Z, +,\cdot)$, then $p$ is a prime number.
My attempt:
1) $(p)$ is a maximal ideal, so it is a prime ...
3
votes
0answers
545 views
How to compute the integral closure of $\Bbb{Z}$ in $\mathbb Q(\sqrt[n]{p})$?
We have the definition of integral closure that all the integral elements of A in B. Could we just compute the integral closure of certain A in B. I am considering such a problem that given a prime p, ...
2
votes
1answer
212 views
Use of GCD when solving linear equations in a ring of integers
I know that when I'm solving a linear equation of the form ax = b (mod n), the gcd(a,n) tells me how many solutions there are. I don't really understand why this is. More than a proof, I'm interested ...
5
votes
1answer
459 views
Idea behind the definition of different ideal
Let $L/K$ be an extension of number fields. Let $I$ be a fractional ideal in $L$ and $$I^*:=\{x\in L \mid \text{Tr}_{L/K}(xI)\subset \mathcal{O}_K\}.$$
The different of $I$ is the following fractional ...
2
votes
1answer
63 views
Conway's proof of the Euclid lemma
Sorry for a potentially very stupid question, but I've stuck in the very beginning of the book "On Quaternions and Octonions" by Conway and Smith with a proof of the well-known lemma:
If $p$ is a ...
5
votes
1answer
287 views
Prove that a specific ring of integers is not monogenic
I'm trying to prove that the ring of integers of $K=\mathbb{Q}(\sqrt7, \sqrt13)$ is not of the form $ \mathbb {Z}[a]$ for some $a$.
Unfortunately I can not figure out where to start. I tried to ...
2
votes
0answers
92 views
Who first used the notation $\mathcal{O}_K$ for ring of integers?
I think this is a standard notation since almost every author uses it, but who came up with the notation? After all, what does $\mathcal{O}$ in $\mathcal{O}_K$ stand for? Thanks in advance.
7
votes
1answer
276 views
Show that the ring of integers $A$ of the cubic field $\mathbb Q[x]$ with $x^3=2$ is principal.
Show that the ring of integers $A$ of the cubic field $K=\mathbb Q[x]$ with $x^3=2$ is principal.
The hint given in the book is to majorize the discriminant of $A$ by $D(1,x,x^2)$ and then use the ...
2
votes
2answers
86 views
If $K:=\mathbb Q\left(\sqrt{-3}\right)$ and $R$ is the ring of integers of $K$, then $R^{\times}=\mathbb Z\big/6\mathbb Z$
How to show that if $K:=\mathbb Q\left(\sqrt{-3}\right)$ and $R$ is the ring of integers of $K$, then the group of units $R^{\times}=\mathbb Z\big/6\mathbb Z$
Now since $-3\equiv1\mod 4$ the ring of ...
3
votes
1answer
83 views
Least rational prime which is composite in $\mathbb{Z}[\alpha]$?
Sébastien Palcoux asked if there was some irrational algebraic $\alpha$ such that all rational primes are primes in $\mathbb{Z}[\alpha].$ MooS answered that there are no such $\alpha.$ This leads to a ...
9
votes
2answers
7k views
$\mathbb Z[\sqrt{-5}]$ is not a UFD [duplicate]
Prove that the ring of integers of $\mathbb Q (\sqrt{-5})$ does not have unique factorisation.
Since $-5\equiv 3\pmod 4$, I know that the ring of integers of $\mathbb Q (\sqrt{-5})$ is $\mathbb Z [\...
6
votes
2answers
173 views
Is $\mathcal{O}_K$ always isomorphic to $\mathbb{Z}[X]/(f(x))$, for some irreducible polynomial $f(x)$?
Given an algebraic number field $K$ and its ring of integers $\mathcal{O}_K$, is $\mathcal{O}_K$ always isomorphic to $\mathbb{Z}[X]/(f(x))$, for some irreducible polynomial $f(x)$?
Since $\mathcal{O}...
0
votes
1answer
50 views
Powers of complexes modulo a prime $p$
We have, for a residue number system, $a^{n+(p-1)} \equiv a^n \bmod p$. In other words, the powers of $a$ repeat after $p-1$ iterations.
We can work with complex numbers by representing a number $$n ...
1
vote
1answer
252 views
Set of algebraic integers is closed under addition and multiplication
If $\alpha$ and $\beta$ are algebraic integers, then show $\alpha + \beta$ and $\alpha \times \beta$ are both algebraic integers.
I know that an algebraic integer is a root of some monic polynomial ...
3
votes
1answer
319 views
Algorithms for finding the ring of integers
In the book's Algebraic Number theory, Ian StewarT, Third edition (page 51-52), has the following propositions:
Theorem 2.20: Let $G$ be an additive subgroup of $\mathfrak{O}_K$ of rank equal to the ...
1
vote
1answer
89 views
Detail in Theorem 12 pag 33, from Marcus book “Number Field”
Let $K, L$ be number fields (i.e. subfields of $\mathbb C$ of finite degree over $\mathbb Q$) of degree $m, n$ over $\mathbb Q$ respectively and assume $[KL:\mathbb Q]=nm$.
Consider $KL$ to be the ...
0
votes
1answer
213 views
Dedekind rings which are UFDs but not PIDs?
I just have a really quick question of an example that I was trying to come up with.
Are there any number rings which are UFDs but not PIDs?
5
votes
1answer
203 views
Non unique factorization domains with prime factorizations with differing number of primes
As is well-known, $Z[\sqrt{-5}]$ is not a ufd because $6$ has more than one prime factorization in this ring: $6=2\cdot 3$ and $6=(1+\sqrt{-5})(1-\sqrt{-5})$. But both of these prime factorizations ...
4
votes
3answers
186 views
Is $\mathcal{O}_{\mathbb{Q}(\sqrt{5})} = \mathbb{Z}[\phi]$?
Is $\mathcal{O}_{\mathbb{Q}(\sqrt{5})} = \mathbb{Z}[\phi]$, where $\phi={1+\sqrt{5}\over 2}$ is the golden ratio?
I know that $5 \equiv 1 \mod 4$, so that then $\mathbb{Z}[\sqrt{5}]$ is not closed as ...
3
votes
3answers
384 views
Ring of integers in a cubic extension
Let $L=\mathbb{Q}[\alpha]$, with $\alpha^3=10$. How can be proved that
$$\frac{\alpha^2+\alpha+1}{3}$$
is in $O_L$, the ring of integers of $L$?
2
votes
1answer
503 views
The ring of integers of the composite of two fields
Let $K,L$ be two number fields and let $KL$ denote the composite field (the smallest subfield of $\mathbb{C}$ containing both $K$ and $L$). Denote respectively by $R,S$ and $T$ the ring of algebraic ...
3
votes
1answer
229 views
Ring of integers of $K=\Bbb Q[u]$ where $u=\sqrt[3]{p^2q}$
Let $p,q$ be distinct prime numbers $\ge 5$ such that $pq^2 \not\equiv 1\mod9$. Let $K=\Bbb Q[u]$ where $u=\sqrt[3]{p^2q}$, and $A$ be the ring of integers of $K$. I have shown that $u,v=pqu^{-1}\in A$...
