# Is the degree of an irreducible inseparable polynomial always a p-power?

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.

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 $$p$$!