Determine the degree of the splitting field of the following:
[tex]x^6 + 1[/tex]
A splitting field is the smallest field that contains all of the root of the polynomial.
The Attempt at a Solution
I got the roots, [tex] +- i, +-\omega_1 and +-\omega_2[/tex]
So the degree of the thing has to have all of the rationals as well as i and those other 2, which are complex numbers.
So here's my confusion:
Does the polynomial have to be irreducible? Do I have to find the irreducible poly to find the degree of the splitting field?
This polynomial *looks*(not that what it looks like MEANS anything) like it's degree 6, so then the splitting field would be 6*2 where the 2 comes from theh degree of the complexes over the rationals? We only did one example in class, and I'm not really clear on that. I'm REALLY REALLY lost on this.
Any input will be appreciated