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.

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27 views

Second summand to make projective module free

Suppose there's a projective $R$-module $P$ (non-free). We know that there is another $R$-module $M$ such that $P\oplus M$ is free over $R$. Is there a way to write down such an $M$ in terms of $P$? ...
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polynomial multiplication with integer polynomial result

A(z), B(s), D(s) are polynomials. A(s) is a polynomial ...
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1answer
213 views

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{...
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1answer
111 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 $...
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1answer
60 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)= \...
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1answer
83 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]...
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1answer
154 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 $f \in \mathbb{Z}[x]$ ...
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77 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 ...
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2answers
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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 ...
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1answer
85 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 ...
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2answers
40 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 ...
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1answer
77 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}[\...
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1answer
58 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{...
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5answers
161 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
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1answer
112 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 $\mathbb{...
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$\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}...
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3answers
112 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 ...
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4answers
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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 ...
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1answer
648 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 $L$ ...
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1answer
34 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 ...
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1answer
45 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'...
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2answers
63 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 $\...
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1answer
287 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 ...
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1answer
307 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
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1answer
50 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 ...
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94 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
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168 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
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1answer
174 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 ...
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2answers
606 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 ...
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1answer
245 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 ...
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4answers
81 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 ...
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685 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, ...
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1answer
246 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
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1answer
574 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 ...
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1answer
66 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
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1answer
353 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 ...
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0answers
106 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.
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1answer
355 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
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2answers
90 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
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1answer
85 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 ...
12
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2answers
10k 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
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2answers
207 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}...
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1answer
56 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
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1answer
432 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
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1answer
407 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 ...
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1answer
104 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 ...
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1answer
230 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
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1answer
250 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
210 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
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3answers
450 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 $L$?