# 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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### 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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### 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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### 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 $\Bbb Q(\sqrt{2})$

I know$^{(1)}$ that the ring of integers of $K=\Bbb Q(\sqrt{2})$ is $\Bbb Z[\sqrt{2}]$ 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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### 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 ...