I can deduce from the "Tower theorem" (for extensions of fields) that if [K:F] is prime then there is no field E such that $\displaystyle F \subset E \subset K$. I can't see how this relates to the last part of the question - how do we go about it?

"Assuming that fields of order 32 exist, show that there are exactly 6 irreducible polynomials of degree 5 in the ring $\displaystyle Z_2[x]$"