What is the splitting field of the polynomial $x^ {a^n}$ −1 in \ the ring Z/aZ[x] with n natural?
I´m working in some kind of proof of the best known theorem that says it´s impossible there exist a finite algebraic field, and in this way, I found this question
(First of all, the a must be prime for it have sense to be Z/aZ[x] field)
As you observed, for $\Bbb{Z}/a\Bbb{Z}$ to be a field, it is necessary and sufficient that $a$ is a prime number. But in that case, by the so called Freshman's Dream: $$x^{a^n}-1=(x-1)^{a^n}.$$ This means that the splitting is the field $\Bbb{Z}/a\Bbb{Z}$ as $x=1$ is the only zero (with multiplicity $a^n$), so your polynomial splits into linear factors over the prime field.