Mathematics Stack Exchange
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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.

In algebraic number theory, the ring of integers $\mathcal{O}_K$ of a number field $K$ is the ring of all elements of $K$ which are roots of a monic polynomial with rational integer coefficients. This tag is often used with 'algebraic-number-theory'.

68 questions
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Easy way to show that $\mathbb{Z}[\sqrt[3]{2}]$ is the ring of integers of $\mathbb{Q}[\sqrt[3]{2}]$

This seems to be one of those tricky examples. I only know one proof which is quite complicated and follows by localizing $\mathbb{Z}[\sqrt[3]{2}]$ at different primes and then showing it's a DVR. Does anyone know any simple quick proof?
pki
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$\mathbb Z[\sqrt{-5}]$ is not a UFD

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 $\mathbb Z [\sqrt{-5}]$. I assume the way to prove that this…
AccioHogwarts
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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 finite extension of $\mathbb{Q}$, then…
Rubertos
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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 quickly the discriminant…
Watson
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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 $L$ are linearly disjoint, but is this also…
Servaes
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Find an integral basis of $\mathbb{Q}(\alpha)$ where $\alpha^3-\alpha-4=0$

Let $K=\mathbb{Q}(\alpha)$ where $\alpha$ has minimal polynomial $X^3-X-4$. Find an integral basis for $K$. I have calculated the discriminant of the minimal polynomial is $-2^2 \times 107$, so the ring of algebraic integers is contained in…
Spook
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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 $\mathbb{Z}\left[\frac{1+\sqrt{5}}{2}\right]$. I would…
Tomiri
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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 $\mathbb{Z}[{\sqrt{5}}]$? I know that it will only…
Mathejunior
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On determining the ring of integers of a cubic number field generated by a root of $x^3-x+1$

I have the following question: Let $\alpha$ be a root of the polynomial $f(x) = x^3-x+1$, and let $K = \mathbb{Q}(\alpha)$. Show that $\mathcal{O}_{K} = \mathbb{Z}[\alpha]$. As I understand it, I need to show that $\{1, \alpha, \alpha^{2}\}$ form…
PFHayes
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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 accurate factorizations into irreductible (not…
Loic
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Algebraic integers of a cubic extension

Apparently this should be a straightforward / standard homework problem, but I'm having trouble figuring it out. Let $D$ be a square-free integer not divisible by $3$. Let $\theta = \sqrt[3]{D}$, $K = \mathbb{Q}(\theta)$. Let $\mathcal{O}_K$ be the…
Zhen Lin
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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 ideal $$\mathcal{D}_{L/K}(I):=(I^*)^{-1}.$$ I…
user72870
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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 fact that every ideal class of a number field $K$ of…
odnerpmocon
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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 $\mathcal{O}_K/\mathcal{m}$ is a finite field for any…
baltazar
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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 have the same number $(2)$ of prime factors...Am I…
Richard Peterson
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