Exercise
The polynomial f = X2 + X + 1 is a divisor of X15 - 1 in Z/2Z[X].
How many zeros does f have in a field of order 8?
0
1
2
No. The polynomial f divides X16 - X,
which has 16 solutions in the field, so ...
No. The polynomial f divides X16 - X,
which has 16 solutions in the field, so ...
Yes! For
f divides X16 - X,
which has 16 solutions in the field, and so is a product of 16 linear factors,
whence X2 + X + 1 is a product of two linear factors.