Determining A Splitting Field

Determining A Splitting Field

I am trying to determine the splitting fields of a bunch of polynomials.
I'll ask one here and hope that a general enough technique can be
described to find the rest of them.
Currently, I'm trying to find the splitting field of
$(x^{15}-5)(x^{77}-1)$ over $\Bbb Q$, find the degree, and determine if
it's a Galois extension.
Now, I know that the right polynomial is the cyclotomic polynomial, hence
has degree $\varphi(77)=60$, over $\Bbb Q$. The left polynomial is
irreducible by Eisenstein's Criterion, hence adjoining $\sqrt[15]{5}$
gives a degree 15 extension and as a separate extension, adjoining
$\zeta_{15}$ (a primitive $15^{th}$ root of unity) gives a degree $8$
extension. All of this seems well and good, but now I'm lost. The
splitting field itself is obvious, but how can I check the degree and
determine if it is Galois?