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Number Theory Integral closure of rational polynomial ring


Nov 20, 2012
Consider the ring $\mathbb{Q}[X]$ of polynomials in $X$ with coefficients in the field of rational numbers. Consider the quotient field $\mathbb{Q}(X)$ and let $K$ be the finite extension of $\mathbb{Q}(X)$ given by $K:=\mathbb{Q}(X)[Y]$, where $Y^2-X=0$.

Let $O_{K}$ be the integral closure of $Q[X]$ in $K$. Certainly $O_{K}$ contains both $\mathbb{Q}[X]$ and $Y$, hence
$$ O_{K}\supseteq \mathbb{Q}[X][Y]$$
My guess is that actually "=" holds. How can be proved this?
Thank u all in advance
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