F field If fx element of Fx can be written as the product of

F field. If f(x) element of F[x] can be written as the product of two polynomials of lower degree, prove that f(x) is reducible in F[x].

Solution

Let f(x) F[x] = p(x) q(x) where p(x) and q(x) f[x] are both polynomials in F[x] of degree less than that of f(x). We know that an irreducible polynomial is a non-constant polynomial that cannot be factored into the product of two non-constant polynomials. Since f(x) can be factored into the factors p(x) and q(x), therefore, by definition itself, f(x) is reducible in F[x].