Example

Suppose that K is a field of order 4. Then L = {0,1} is a subfield of order 2. Take y K\L. The theorem tells us that K is a 2-dimensional vector space over L, and so 1, y is a basis of K over L. In particular, there are a, b L such that y² = a + by.

Now consider the linear transformation x -> yx of K. It has matrix

0 a 1 b

As y must be invertible, we have a 0. But then a = 1.

There remain two possibilities for b. Suppose b = 0. Then y² = 1. But from this we deduce (y + 1)² = 0, and so y + 1 = 0, that is, y = 1, a contradiction. Hence b = 1, and so y satisfies y² = y + 1.

In conclusion, K = {0, 1, y, y + 1} with the multiplication rule y² = y + 1.

The above argument gives a glimpse of why there is just one field of order 4. Here is another way of interpreting the result. The element y is a zero of the irreducible polynomial X²+X+1. Thus, it behaves in the same way as the residue of X in the field L[X]/(X² + X + 1). In fact, K is isomorphic with this field.