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Let $\mathbb{F}_{p}(\alpha)/\mathbb{F}_{p}$ be a transcendent field extension and $f(x)=x^{p}+\alpha$. Then could it be that $f$ is irreducible over $\mathbb{F}_{p}(\alpha)$? And why or why not?

I assume that it is.

Celsius
  • 208
  • I know how to show that $f$ is irreducible yet.

    Let $K=\mathbb{F}_{p}$. Then the transcendency of $\alpha$ provides a canonic isomorphism between $K[\alpha]$ and the polynomial ring $K[Y]$. With $deg(Y)=1$ it follows that $Y$ is irreducible in $K[Y]$ and hence prime. That implies $\alpha$ is prime in $K[\alpha]$ and we can use Eisenstein's criterion which says that $f$ is irreducible in $K[\alpha][X]$ and thereby in $K(\alpha)[X]$.

     – Celsius May 06 '14 at 18:39

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