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## Homework Statement

Let [tex]p\in\mathbb{Q}[x][/tex] be an irreducible polynomial. Suppose K is an extension field of [tex]\mathbb{Q}[/tex] that contains a root [tex]\alpha[/tex] of p such that [tex]p(\alpha^2)=0[/tex]. Prove that p splits in K[x].

## The Attempt at a Solution

I was thinking contradiction, but if p does not split in K, the only logical conclusion I can come to is that there is an extension field L[x] such that p splits and [tex]K[x] \subseteq L[x][/tex].