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Why is $\overline{\mathbb{F}_p}((t))$ transcendental over $\mathbb{F}_p((t))$ ?

I guess $\overline{\mathbb{F}_p}((t))$ is not unramified over $\mathbb{F}_p((t))$ because $\overline{\mathbb{F}_p}((t))$ is transcendental over $\mathbb{F}_p((t))$・・・①.

But why ① holds? I couldn't check ① by myself. What element is transcendental element? And why? (In general, to prove transcendentality exactly is difficult, is it managable in this case?)

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    $\begingroup$ Consider $\sum_{n = 0}^\infty c_n t^n$, where $c_n$ generates $\mathbb F_{p^n}$. $\endgroup$
    – LSpice
    Commented Sep 13, 2021 at 16:54
  • $\begingroup$ How to prove the element is transcendental ? $\endgroup$
    – Duality
    Commented Sep 15, 2021 at 5:27

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