The following is an extract from my book
I'm having trouble understanding the very last part of the proof, namely, why can't the terms of highest degree cancel?
2026-04-08 05:15:36.1775625336
Let $u$ be transcendental over $\mathbb{Z}_p$. Why is $t^p-u$ irreducible over $\mathbb{Z}_p(u)$?
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The top degree term of $v(u)$ has degree divisble by $p$. The top degree term of $uw(u)^p$ has degree congruent to $1$ mod $p$ (the $w(u)^p$'s leading term has degree divisible by $p$ and multiplying by $u$ adds $1$) and in particular not divisible by $p$. So these two terms are unequal and hence cannot cancel.