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Consider a field $K$ of characteristic $p>0$. Let $f$ be an irreducible, inseparable polynomial in $K[Y]$. I'm wondering if the degree of $f$ has to be a power of $p$.

For instance, the standard example is $f=Y-X^p\in \mathbb{F}_p(X)[Y]$ and it factors as $f=(\sqrt[p]{Y}-X)^p$. All examples that I can find are in essentially coming from this one. I can't thing about anything else.

In particular I'm asking my self if it is possible that: if $f$ is an inseparable irreducible polynomial in $K[Y]$, f factors in $L=K[Y]/f$ as $$(*)\;\;\;\;\;f=g\cdot\prod_{i=1}^d(Y-\alpha_i),$$ with the $\alpha_i\in L$ pairwise distinct and $g$ irreducible in $L[Y]$ and inseparable over $L$.

I don't think so, but I can't see it properly. Also I can't find a precise statement in the literature.

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Up to a multiplicative constant $c\in K^*$, an irreducible polynomial $f=\sum_{i=0}^n c_i X^i\in K[X]$ is the minimal polynomial $f_K^\alpha$ of some $\alpha\in \overline{K}$. Since we must have $f'=0$ in $K[X]$, we see by using the formal derivative that $f=\sum_{i=0}^n a_iX^{ip}$.

Hence $\deg f$ is always a multiple of $p$!

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