Prove a certain cyclic extension with prime power order is simple
Solution 1:
Since E/F is cyclic of degree $p^n$, for any integer m less or equal to n, there is a unique subextension $F_m$/F of degree $p^m$ , and E is the union of the ascending chain of the $F_m$ ' s . Your hypothesis is that E = $F_{n-1}$(a) . If F(a) were not equal to E, a would belong to a certain $F_m$ with m < n , hence a would be in $F_{n-1}$ : contradiction .