Articles of integral dependence

Integral extension (Exercise 4.9, M. Reid, Undergraduate Commutative Algebra)

Let $k$ be any field and let $A = k[X,Y,Z]/(X^2 – Y^3 – 1, XZ – 1)$. How can I find $\alpha, \beta \in k$ such that $A$ is integral over $B = k[X + \alpha Y + \beta Z]$? For these values of $\alpha$ and $\beta$, how can I find concrete generators for $A$ […]

Extending an automorphism to the integral closure

I need some help to solve the second part of this problem. Also I will appreciate corrections about my solution to the first part. The problem is the following. Let $\sigma$ be an automorphism of the integral domain $R$. Show that $\sigma$ extends in an unique way to the integral closure of $R$ in it’s […]

How to compute the integral closure of $\Bbb{Z}$ in $\mathbb Q(\sqrt{p})$?

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

Algebraic vs. Integral Closure of a Ring

Let $R\subseteq S$ be a ring extension. It is true that the set of elements of $S$ that are are integral over $R$ (i.e. satisfy a monic polynomial equation over $R$) is a subring of $S$. Can anyone provide an example showing that the set of elements of $S$ that are merely algebraic over $R$ […]

Hints about Exercise 4.2 in Miles Reid, Undergraduate Commutative Algebra

$ A \subset B $ is a ring extension. Let $ y, z \in B $ elements which satisfy quadratic integral dependance $ y^2+ay+b=0 $ and $ z^2+cz+d=0 $ over $ A $. Find explicit integral dependance relations for $ y+z $ and $ yz $. It is given this hint start by assuming $1/2 […]

$\mathbb{Q}/(X^2+Y^2-1)$ is integrally closed

I’m was browsing this question, where it is proven the quotient field of $\mathbb{Q}[X,Y]/(X^2+Y^2-1)$ is isomorphic to the rational function field $\mathbb{Q}(t)$ under the isomorphism $$ (x,y) \mapsto \left( \frac{2t}{t^2+1}, \frac{t^2-1}{t^2+1} \right). $$ How can this isomorphism be used to show $\mathbb{Q}[X,Y]/(X^2+Y^2-1)$ is normal in its quotient field? I identified $\mathbb{Q}[X,Y]/(X^2+Y^2-1)$ with $\mathbb{Q}\left[\frac{2t}{t^2+1}, \frac{t^2-1}{t^2+1} \right]$, […]

Normalisation of $k/(y^2-x^2(x-1))$

I am trying to figure out the normalisation of $k[x,y]/(y^2-x^2(x-1))$, for an algebraically closed field $k$. I can show that it is not normal and I have the information that the normalisation is $k[t]$, where $t=\frac{y}{x}$. I can’t figure out how to prove that this is the normalisation i.e. that the fraction fields of $k[x,y]/(y^2-x^2(x-1))$ […]

Working out the normalization of $\mathbb C/(X^2-Y^3)$

I’m trying to identify the normalization of the ring $A := \mathbb C[X,Y]/\langle X^2-Y^3 \rangle$ with something more concrete. First, $X^2-Y^3$ is irreducible in $\mathbb C[X,Y]$, making $\langle X^2-Y^3\rangle$ prime, so $A$ is a domain and it makes sense to talk about its normalisation, i.e., its integral closure in $\mathrm{Frac}(A)$. Then, we try to understand […]

Easy way to show that $\mathbb{Z}{2}]$ is the ring of integers of $\mathbb{Q}{2}]$

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?

Normalization of a quotient ring of polynomial rings (Reid, Exercise 4.6)

I solved all parts of Exercise 4.6 of the book Undergraduate Commutative Algebra of Miles Reid except the last one. Let $A=k[X]$ and $f\in A$ has a square factor but it is not a square polynomial itself, then what will be the normalization of $B=k[X,Y]/(Y^2-f)$ in terms of $f$? What should I do for this […]