Discussion

The number of non-constant irreducible polynomials of degree \(\le 3\) in \(\mathbf{Z}_{2}[x]\) is :

Answer: A

There are 5 such polynomials.

Count by degree over \(\mathbf{Z}_{2}\):

Degree 1: \(x\) and \(x + 1\), both irreducible. That is 2.
Degree 2: a quadratic is irreducible exactly when it has no root in \(\mathbf{Z}_{2}\). Of \(x^{2}, x^{2} + 1, x^{2} + x, x^{2} + x + 1\), only \(x^{2} + x + 1\) has no root (\(x^{2} + 1 = (x + 1)^{2}\)). That is 1.
Degree 3: again irreducible exactly when there is no root. The constant term must be 1 (else 0 is a root) and the number of terms must be odd (else 1 is a root), leaving \(x^{3} + x + 1\) and \(x^{3} + x^{2} + 1\). That is 2.

Total \(2 + 1 + 2 = 5\). The count 3 forgets the linear polynomials, which are always irreducible, and 4 or 6 come from miscounting the cubics.

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