Splitting field for polynomial

Question

How can I find the splitting field for the polynomial $$x^{p^{50}}-1$$ ??

Could you give me some hints??

To find the splitting field for a polynomial, we have to find all the roots of this polynomial, right?? How could we do this in this case??

Answer 1

Look up cyclotomic polynomials. Those are the irreducible factors of polynomials of the form $x^n-1$ in $\Bbb{Q}[x]$. So if you are working over $\Bbb{Q}$ the splitting field will be $F=\Bbb{Q}(\zeta)$ with $\zeta=e^{2\pi i/p^{50}}$. Assuming that $p$ is a prime its degree is $[F:\Bbb{Q}]=\phi(p^{50})=(p-1)p^{49}$.

If you are working over a field other than $\Bbb{Q}$, e.g. one of positive characteristic, then the above degree formula no longer applies.