Find all the irreducible polynomials of degree 1, 2, or 4 over , and verify that their product is .

Note that there are (monic) polynomials of degree , as each non-leading coefficient can be either 0 or 1.

The polynomials of degree 1, and , are both irreducible.

There are 4 polynomials of degree 2, one of which is irreducible.

- is reducible
- is reducible
- is reducible
- has no roots, and so is irreducible.

Before we address the degree 4 polynomials, we prove a lemma.

Lemma 1: If is a degree 4 polynomial over with constant term 1 and factors as a product of quadratics, then the linear and cubic terms of are equal. Proof: We have (as we assume) . Now , so that . So , as desired.

Lemma 2: is irreducible over . Proof: Clearly this has no roots. By Lemma 1, if factors into two quadratics, then the factors’ linear terms, and , satisfy , which is impossible over .

There are 16 polynomials of degree 4.

- is reducible
- is reducible
- is reducible
- is reducible
- is reducible
- clearly has no roots, and by the lemma, has no quadratic factors. So is irreducible.
- is reducible
- clearly has no roots, and by Lemma 1, has no quadratic factors. So is irreducible.
- is reducible
- is reducible
- is reducible
- has 1 as a root, so is reducible
- is reducible
- has 1 as a root, so is reducible
- is reducible
- is irreducible

So there are six irreducible polynomials of degree 1, 2, or 4 over :

It is easy (if tedious) to verify that the product of these polynomials is . (WolframAlpha agrees.)