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Are these two fields $\mathbb {F}_{p}(x)$ and $\mathbb {F}_{p}(x^p)$ are isomorphic fields?

Only thing I know there exists embeddings from one field to another and the extension $\mathbb {F}_{p}(x)/\mathbb {F}_{p}(x^p)$ is Galois with Galois group cyclic of order $p$. I don’t know if I have to really use these information. Help me. Thanks.

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  • $\begingroup$ This extension is not Galois, because the extension is not seperable. The minimal polynomial of $x$ in $\mathbb{F}_p(x^p)$ is $(t-x)^p = t^p-x^p$. $\endgroup$
    – Guenterino
    Commented Jul 19, 2021 at 6:43

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$\mathbb{F}_p(x)$ and $\mathbb{F}_p(x^p)$ are both purely transcendental extensions of $\mathbb{F}_p$ and they both have transcendence degree $1$ over $\mathbb{F}_p$. Because purely transcendental extensions of the same transcendence degree are isomorphic, these two are isomorphic (see for example here page 2, shortly after Definition 12.3).

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