I want to prove that: the integral domain $k[x,y]/ (x^3 - y^2)$ is not normal and find its integral closure in its field of fractions. I know that I will use the idea that $UFD \implies$ normal and hence not normal means not $UFD$ so what I should show is that $k[x,y]/ (x^3 - y^2)$ is not a UFD and I have read a hint here  to consider $x = (y/x)^2.$but I can not see how that will show that it is not a UFD. could someone elaborate this to me please? Also, what is the field of fraction of this integral domain? I know that a unique factorization domain is an atomic integral domain in which factorization to irreducibles is unique (up to associates). And a commutative ring is atomic if each $r \in R$ is a finite product of irreducibles in R. Also, a ring $R$ is a normal ring if $R$ is an integral domain that is integrally closed in $\operatorname{frac}(R).$ And $R$ is integrally closed in a commutative ring $S$ iff the integral closure of $R$ in $S$ (int($R$)) equals $R$. And int($R$) consists of all $x \in S$ that is integral over R. And note that an element is integral over $R$ if there exists a monic polynomial $f(x) \in R[x]$ such that $f(x) = 0.$
I'm trying to show that $R = \mathbb{Q}[X,Y]/(Y^2-X^3)$ is not a UFD, but I got stuck.  To prove this, I could try to find two "different" factorisations for one element, but I am not familiar with this, so I tried to use a lemma and one of the previous exercises. If my syllabus is right, in every UFD counts $$ x \ \text{is irreducible} \quad \iff \quad x \ \text{is prime}$$ My syllabus also states that the elements $\bar{X}, \bar{Y} \in R$ are irreducible. So I tried to show that at least one of the elements $\bar{X}, \bar{Y} \in R$ does not generate a prime ideal.  This would mean that I should find two polynomials $f,g \in \mathbb{Q}[X,Y]$, such that  $$\exists p \in \mathbb{Q}[X,Y], \quad fg - pX \in (Y^2-X^3)$$ and $$\forall q \in \mathbb{Q}[X,Y], \quad f-qX \notin (Y^2-X^3) \ \wedge \ g-qX \notin(Y^2-X^3)$$ or the same thing but then for $Y$.  I hope that you can tell me if this approach is correct, and provide me a hint. I'd appreciate it if you told me the solution, but please start your answer with a hint, clearly separated from the rest.
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This way the ETF price is automatically regulated and the authorised shareholders earn a profit. Everybody is happy. In contrast to a fund, when we buy an ETF we are not generally buying ETF shares from the ETF manager. Instead, we are buying ETF shares from another investor. Therefore... as I understand it, no new shares are really bought. And thus, following the process above, if we buy the ETF we push its price upwards. This again creates a &quot;gap&quot;, this time the price of the ETF is momentarily higher than its underlying assets and, therefore, authorised shareholders will sell ETF shares and buy shares of A, B and C. So... an ETF manager only buys a fixed amount of shares when the ETF is created? I want to know if I am correct in this reasoning about how the basics of physical ETFs work. I have tried to keep things as simple as possible. I understand that there will be lots of quirks and quincks, and it is great if these are elaborated in the answers. 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