Prove that there are no separable extensions of $k$ of degree $n$

Question

Let $k$ be a field and let $n \gt 0$ be an integer. Assume that there are no irreducible polynomials of degree $n$ in $k[x]$ . Prove that there are no separable extensions of $k$ of degree $n$

I don't understand this question. There cannot be any extension of degree $n$, rest alone separable. Is it not??

Am I missing something here??

Thanks for the help!!

Answer 1

You are missing the primitive element theorem. That is the critical argument.

To be more precise: a separable extension of degree $n$ would have a primitive element, whose minimal polynomial would be irreducible of degree $n$, contradiction!