determine the splitting field K/Q and calculate [K:Q]
i)
I think i understand just need reasurance
spliting field
[K:Q] =4 because is min polynomial
But you haven't yet determined , and its degree is wrong: certainly , but this extension field of the rationals is NOT
the splitting field of the given pol. over the rationals since it doesn't contain ALL its roots...for example, is a root of the pol. that isn't contained in .
What you did is just fine to find the roots: , but then it must be clear that both roots of the rightmost factor are complex
non-real ( in fact, a conjugate pair: ) , so they aren't contained in a real field as .
Take it from here and try now to find explicitly what is and its degree over the rationals.
2)
splitting field
[K:Q] = 4
is this all correct?
This time the degree is correct (though you gave no explanation at all... ) but again you haven't explicitly described the splitting field ...