Proving that every element of a coordinate ring can be written uniquely in the form $ P(X) + Q(x)y $

Question

If $ X \subset \mathbb{A}^{2} $ be the algebraic plane curve defined by $ y^{2} - x^{3}. $ Prove that an element of $ k[X] $ can be written uniquely in the form $ P(x) + Q(x)y $ with $ P(x), Q(x) $ polynomials.

Now, $$ k[X] = k[x,y]/\Big(y^{2}-x^{3}\Big) = \Big\lbrace f(x,y)+g(x,y)\Big(y^{2}-x^{3}\Big)\Big\rbrace , $$ for all $ f(x,y),g(x,y) \in k[x,y]. $

I'm unsure how to proceed from here.

Answer 1

As a module over $k[x,y^2]$, we have $k[x,y]\cong k[x,y^2]\oplus yk[x,y^2]$. Let $f(x,y)=g(x,y^2)+yh(x,y^2)$. Then in $k[x,y]/(y^2-x^3)$, $f(x,y)$ has the same image as $g(x,x^3)+yh(x,x^3)$. Now, just note that $g(x,x^3)$ and $h(x,x^3)$ are both polynomials in $x$.

Note, this follows from a mildly souped up version of the fact that for any single variable polynomial $P(x)$, $P(a)-P(b)$ is divisible by $(a-b)$, which in turn follows from the identity $a^n-b^n=(a-b)\displaystyle\sum_{i+j=n-1}a^ib^j$.

Answer 2

Well, the standard basis of $K[x,y]/\langle y^2-x^3\rangle$ is given by $1,x,y,xy,x^2,x^3,x^4,\ldots$. Thus each element of the quotient ring has the form $f(x) +yg(x)$.