Exercise
Let F be a field containing the q = pn elements 0 = a0, ... , aq-1. Consider the following two statements.
A. We have the factorization
Xq-1-1 = (X-a1)
··· (X-aq-1).
B. Substituting X = 0 gives
-1 = a1a2 ···
aq-1 (mod p).
Which are correct?
Both. |
Neither.
|
|
Only A. |
Only B.
|
That is right. Try to apply this to the field Z/pZ, what do you get?
Wrong.
According to
the theorem for each nonzero element we have
Xq-1 = 1, hence it is a root of
Xq-1-1.
You are wrong. Although you have to make a careful distinction between a polynomial and the related polynomial function, in this case it is correct.