3 (Irreducible polynomials) A polynomial f ? F s called irreducible if deg f 1 and f has no factors other than constants in F, and associates of itself. In other words, it is not possible to write f = gh for g, h ? F[x] satisfying deg g, deg h > 1. An irreducible polynomial can be thought of as an analog of a prime integer. A polynomial that is not irreducible is called reducible (a) Explain why the polynomial a2-2 is reducible in R[z], but irreducible in Q[x] (Notice that it is an element of both of these rings!) (c) If deg f = 2 or 3, then f is reducible if and only it has a root (ie., f(c) = 0 for some constant c ? F): If f = gh for polynomials g and h of degree at least 1, then deg f = degg + degh, so 2 degg + deg h
