Algebraic Elements and Field Extensions

Please can someone help me with this? Even just pointers would be greatly appreciated.

Which bits are you having trouble with?

For the first part I can see that by the [Q(α):Q]<∞ it is true, but I dont know how to write this rigorously.

(b) I am part way through and am happy to leave to my tutors to explain thoroughly

but (c) I really don't know where to start.