I've been boggling over where to go for these for a bit, having some trouble thinking of the right direction.
1.) Find (F:Q) (i.e. the degree of F over Q) if F is the splitting field for the polynomial x^5 - 1.
the roots of x^5 - 1 are x = 1, and the four other roots of unity. A splitting field is the smallest extension of Q containing the roots. I had thought the degree was 5, but I've been told that's incorrect...
2.) Let F be a number field and let a,b exist in C such that F(a) and F(b) are finite extensions of F. Prove (F(a,b):F(a)) is less than or equal to (F(b):F).
From Tower Law, (F(a,b):F) = (F(a,b):F(a))(F(a):F) = (F(a,b):F(b))(F(b):F). Not quite so sure how to manipulate this correctly.
1.) Find (F:Q) (i.e. the degree of F over Q) if F is the splitting field for the polynomial x^5 - 1.
the roots of x^5 - 1 are x = 1, and the four other roots of unity. A splitting field is the smallest extension of Q containing the roots. I had thought the degree was 5, but I've been told that's incorrect...
2.) Let F be a number field and let a,b exist in C such that F(a) and F(b) are finite extensions of F. Prove (F(a,b):F(a)) is less than or equal to (F(b):F).
From Tower Law, (F(a,b):F) = (F(a,b):F(a))(F(a):F) = (F(a,b):F(b))(F(b):F). Not quite so sure how to manipulate this correctly.
