find all irreducible polynomials of degree 2 in z2 find all

Solution

A polynomial of degree 2 over Z2 is of the form p(x) = x ² + ax + b.

If b = 0, then p(x) = x ² + ax = x(x + a) or, equivalently, 0 is a root, and p(x) is not irreducible.

Hence to be irreducible, p(x) must be of the form p(x) = x ² + ax + 1.

Observe that p(0) = 1 and p(1) = 1² + a · 1 + 1 = 1 + a + 1 = a.

If a = 0, then 1 is a root and p(x) = x ² + 1 = (x + 1)² is not irreducible. If a = 1, then p(0) = 1 and p(1) = 1,

hence p(x) = x ² +x + 1 has no root, and is therefore irreducible.

why because

If F is a field and p(x) F[x] with deg p(x) = 2 or 3, then p(x) is irreducible over F if and only if p(x) has no root in F

Hence the only irreducible polynomial over Z2 of degree 2 is x ² + x + 1.