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 integers of $K,L$ and $KL$. Let $m,n$ be the degrees of $K$ and $L$ over the rationals and assume that $[KL:\mathbb{Q}]=mn$. Now, […]

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, what is the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[x]/(x^n–p)$. After some trials, I find the answer […]

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 ring of algebraic integers inside $K$. I need to find explicitly elements generating $\mathcal{O}_K$ as a $\mathbb{Z}$-module. […]

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 natural question: Given some irrational algebraic $\alpha,$ what is the least rational prime $p$ such that $p$ is composite in $\mathbb{Z}[\alpha]$? I’m looking for […]

Let $K=\mathbb{Q}(a)$ where $a^3=d$ where $d\neq 0, \pm 1$ is a square free integer. Show that $\Delta (1, a, a^2)=-27d^2$. By calculating the traces of $\theta, a\theta, a^2\theta$ where $\theta=u+va+wa^2$ with $u,v,w\in \mathbb{Q}$ and the norm of $\theta$, show that the ring of integers $\mathcal{O}_K\subset \frac{1}{3}\mathbb{Z}[a]$. So far I have calculated the the traces of […]

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 a $\mathbb{Z}$-basis for $\mathcal{O}_{K}$, but it is not clear what a good method for that is.

I am stuck with problem 22, chapter 3 in Marcus’ book Number Fields which says: Suppose $\alpha^5=2\alpha+2$. Prove that the ring of integers of $\mathbb Q[\alpha]$ is $\mathbb Z[\alpha]$. Prove the same thing also if $\alpha^5+2\alpha^4=2$. Try: Discriminant for both of them is not square free.

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?

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