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 $A$ […]

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 […]

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 […]

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 $R$ […]

$ 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 […]

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 $\mathbb{Q}\left[\frac{2t}{t^2+1}, \frac{t^2-1}{t^2+1} \right]$, […]

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 $k[x,y]/(y^2-x^2(x-1))$ […]

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 […]

Let $A$ be a UFD, $K$ its field of fractions, and $f$ an element of $A[T]$ a monic polynomial. I’m trying to prove that if $f$ has a root $\alpha \in K$, then in fact $\alpha \in A$. I’m trying to exploit the fact of something about irreducibility, will it help? I havent done anything […]

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?

Intereting Posts

Creating a sequence that does not have an increasing or a decreasing sequence of length 3 from a set with 5 elements
Product Notation for Multiplication in Reverse Order
Given a matrix $A$ with a known Jordan decomposition, what is the Jordan decomposition of $A^2+A+I$?
Eigenfunctions of the Laplacian
If $a=\langle12,5\rangle$ and $b=\langle6,8\rangle$, give orthogonal vectors $u_1$ and $u_2$ that $u_1$ lies on a and $u_1+u_2=b$
Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$
Show that $\mathbb{R}/\mathbb{Z}$ is isomorphic to $\{e^{i\theta} : 0 \le \theta \le 2\pi \}$
What are the last two digits of $77^{17}$?
$A \subseteq B \subseteq C$ with $A \subseteq C$ separable and $pd_{B \otimes_A B}(B) = \infty$
Evaluate $\int\sin(\sin x)~dx$
Function theory: codomain and image, difference between them
Finite subgroups of the multiplicative group of a field are cyclic
Proof that $x \Phi(x) + \Phi'(x) \geq 0$ $\forall x$, where $\Phi$ is the normal CDF
In $R$, $f=g \iff f(x)=g(x), \forall x \in R$
Technique for finding the nth term