Examples
A case where the polynomial has fewer zeros than its degree.
The fundamental theorem of algebra.
Consider g = X2 + 1 
Q[X].
Since there is no element in Q squaring to -1,
there are no zeros of X2 + 1 in Q.
Q[X].
Since there is no element in Q squaring to -1,
there are no zeros of X2 + 1 in Q.
The fundamental theorem of algebra says that every polynomial in
C[X] has a zero.
Equivalently:
every polynomial in C[X] is a product of linear
factors.
We shall give no proof of this fact. One reason is that
it is hard, another that we have given no rigorous treatment of
C anyway.