I was going through the proof of the theorem which states that : "The finite dimensional extension L of K is separable iff the bilinear form $(x,y) = T_{L/K}(xy)$ is non degenerate". The proof given in the book "Algebraic Number fields" by Gerald Janusz to prove the reverse implication involves assuming that L is non separable over K and arriving at a contradiction. While doing so, he states the existence of a subfield $F$ of $L$ containing K such that:
(a) $(L:F) = p^m \neq 1$
(b) for each $x \in L,x^p$ is in $F$
I do not understand how is it that such a field $F$ exists. It will be helpful if the construction of such a field can be explained.