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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Ring of integer for the case D=-19 is not a Euclidean domain [duplicate]
I'm solving the following problem. Unfortunately, I don't know any source of the problem because it's a question from another school.
Let $R=\{a+b\alpha|a, b\in \Bbb{Z}, \alpha=\frac{1+\sqrt{-19}}{2}\}...
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need to calculate A^(-x) in finite fields for inverse dft
Trying to implement to the dft in finite fields (GF(m = 337) with 8th_unit_root = 85 since my data is of size 8) - in the inverted dft defined here: https://en.wikipedia.org/wiki/...
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What does a|b in Integerrings mean? [duplicate]
What does a|b in Integerrings mean ? How to read it ?
Found it in german wikipedia under 'Struktursatz über zyklische Arithmetik':
https://de.wikipedia.org/wiki/Sch%C3%B6nhage-Strassen-Algorithmus
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Integerring: Is there a smart and fast way to calculate the modulo of big numbers (of power > 128 ( basis 2)) stored as an array?
Is there a smart and fast way to calculate the modulo of big numbers (of power > 128 ( basis 2)) stored as an array ? Are there some tricks, or theorems I could look up to ?
I thought about ...
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Calculating $[\mathcal{O}_K : \mathfrak{p}]$ using basis
Given the number field $K = \mathbb{Q}(\sqrt{2}, \sqrt{3})$, the ring of integers $\mathcal{O}_K = \mathbb{Z}[\gamma]$ where $\gamma = \frac{\sqrt{2} + \sqrt{6}}{2}$.
$$\langle 5 \rangle = \langle \...
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Abelian subgroup $(M,+) \subset ( \mathbb{Z}[\alpha],+)$ is of the form $m\mathbb{Z}\oplus(a+b\alpha) \mathbb{Z}$
$\mathbb{Z}[\alpha]$ is the quadratic integer ring associated to the squarefree integer $d$.
Let $m\in M \cap\mathbb{Z}$ and $\beta=a+b\alpha\in M$.
If $\delta=x+y\alpha$, write $y=qb+r$, where $0\le ...
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How do I show that the ring of integers of $\mathbb{Q} (i+\sqrt{2} )$ is a Euclidean domain? [duplicate]
Show that the ring of integers of $\mathbb{Q}(i+\sqrt{2} )$ is a Euclidean domain.
Not sure how to go about doing this. I've tried showing that $\mathbb{Q}(i+\sqrt{2} )$'s norm is a Euclidean ...
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Integer ring of local field is DVR?
Let $K$ be a local field, that is, complete with discrete value, and its residue field is finite. Then, is the integer ring of $K$ a DVR?
For example, p-adic number field's integer ring is p-adic ...
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sieve of Eratosthenes generalization to Dedekind domains or even PID's
I'm Interested in finding irreducibles in Dedekind domains, (and especially integer rings) in an efficient manner.
I've tried to look around a bit but found no papers on this (admittedly my paper ...
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The ring of integers of a cubic number field generated by a root of $x^3-2x-2$
Let $\alpha$ be a root of the polynomial $f(x) = x^3-2x-2$, and let $K = \mathbb{Q}(\alpha)$. Show that $\mathcal{O}_{K} = \mathbb{Z}[\alpha]$.
I know the following
$$\text{disc}(1,\alpha,\alpha^2)=-...
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Integer rings and UFDs in transcendental field extensions of $\mathbb{Q}$
I recently began to study algebraic field extensions of $\mathbb{Q}$ aka number fields and especially the definition of algebraic integers in these fields. Some rings of algebraic integers are unique ...
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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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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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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 $...
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$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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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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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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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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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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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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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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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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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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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 $\...
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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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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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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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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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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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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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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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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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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 $\mathbb{Z}[\zeta_3]$ ...
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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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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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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 ...
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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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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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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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$p$ 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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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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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 ...
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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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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 ...
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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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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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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 ...
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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 ...
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