Hi, I need some help with this question.

Let be a field and .

a. The numbers of bases of for is equal to the order of

b. Prove the order of is .

c. Prove the order of is .

Working:

If is a basis of then there is an invertible matrix with these vectors as columns.

Note that can be any non-zero vector, then can be any vector in that is not on the line spanned by , and same for to . So there are choices for , choices for , choices for , etc.

So the order of is

So this gives me the answer to part b, but how do I work out part a and c?

Thanks for any help you provide.