Example

Consider the polynomial

f = X⁴ + X + 1 Z/2Z[X].

Since it is irreducible, the residue class ring K = Z/2Z[X]/(f) is a field. By Section 4.5 it has order 16.

The map x -> x⁴ is a morphism K -> K. We wish to determine its fixed field M = {x K | x⁴ = x}.

Put y = X + (f). Suppose g = ay³ + by² + cy + d M. Then, using

y⁴ = y + 1,   y⁸ = y² + 1,   y¹² = (y + 1)(y² + 1) = y³ + y² + y + 1,

we find

g⁴ = ay³ + (a + b)y² + (a + c)y + (a + b + c + d).

From g⁴ = g we derive

a = 0,   b = c.

Thus, M = {0, 1, y² + y, y² + y + 1}, a subfield of order 4.