Suppose K is a field of order 32. Then K* is a group of order 31. Each element distinct from 1 in K* has order 31, as its order is a divisor of 31 and distinct from 1, see a previous theorem.
Consider the polynomial f = X31 - 1.
In Z/2Z[X], the polynomial f factors into
| f= | (1 + X)(1 + X2 + X5)(1 + X3 + X5) |
| (1 + X + X2 + X3 + X5) | |
| (1 + X + X2 + X4 + X5) | |
| (1 + X + X3 + X4 + X5) | |
| (1 + X2 + X3 + X4 + X5) |
Let a be an element of K which is a zero of 1 + X + X2 + X3 + X5. Then an elementary calculation shows that a2 is also a zero of this polynomial. In fact, 1 + X2 + X4 + X6 + X10 = 0 mod 1 + X + X2 + X3 + X5.
The five zeros of the polynomial are therefore a, a2, a4, a8, a16.
This result could also have been derived by applying
a previous result with
x -> x2.