Let F be a field and fx subset Fx a polynomial of degree 3 P

Let F be a field and f(x) subset F[x] a polynomial of degree 3. Prove that f(x) is irreducible if and only if f(x) has no root in F. Let N be a normal subgroup of a group G and let G/N be the factor group. Let a subset G be an element of finite order o(a) = n. Show that the order of the element aN subset G/N is divisor of n.

Solution

3. F be a field and f(x) F[x] be a polynomial of degree 3. We will prove that f(x) is reducible over f if and only if f(x) has a zero in F. In other words f(x) is irreducible if and only if f(x) has no root in F.

Proof: Suppose that f(x) = g(x)h(x), where both g(x) and h(x) belong to F[x] and have degrees less than that of f(x). Since, deg f(x) = deg g(x) + deg h(x) and deg f(x) = 3, at least one of g(x) and h(x) has degree 1. Say g(x) = ax + b. Then, clearly –a^–1b is a zero of g(x) and therefore a zero of f(x) as well.

Conversely, suppose that f(a) = 0, where a F. Then, by the Factor Theorem, we know that (x – a) is a factor of f(x) and, therefore, f(x) is reducible over F. (Proved)