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I'm reading page $59$ of Reid's "Undergraduate commutative algebra" book.

In example (ii) it says, $k[x^{2}] \subset k[x]$ is an integral extension.

How do we know this? I mean, in order to show this we must take a polynomial $f(x) \in k[x]$ and show there is a monic polynomial $g(x) \in k[x^{2}]$ such that $g(f(x))=0$, right? Why can we do this?

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    @user6495: Sure thing.2011-05-16

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It's enough to prove that $x$ is integral over $k[x^2]$, which it clearly is, being a root of $T^2-x^2 \in k[x^2][T]$.

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    @Zev, tha$n$ks for fixing it!2011-05-16