Definition: A ring is a set A, equipped with two operations 1. Addition: + : R x R -> R, 2. Multiplication: . : R x R - R. such that the following axioms are satisfied: 1. For all a, be R, atb=bta. 2. For all a, b. cent. at (te) = (atbite 3. There exists an element 0 6 R such that for all o e R, Of a = a. We call 0 the additive identity of . 4. For all a = R there exists an element be R such that a + b = 0. We write & = -a and call -a the additive inverse of a 5. For all a, b, ce Rt, table - a(be). (. For all a, b, cc R, a(b + c) = ob + ac and (5 + o)e = but me. Definition Let R be a ring. We say R is unital if there exists I e R such that lo = al = a for all " E R. We call 1 the multiplicative identity (or unity) of R Definition Let A be a ring. We way R is commutative if ab = be for all o, be R. Definition Let R be a commutative and unital ring. We say 0 7 a e R is a unit in R if there exists be R with ab - 1. Definition Let I be a commmitative and unital ring with at least two elements. We say R is a field if and only if every nonzero element of R is a unit Lot F be a field. We let Fly denote the ring of polynomials with coollicicats in F. Prove that if /(x), s(x) = Ffa] with p(x) # 0, then there exists q(), r() e F[x] with r(x) = 0 or dug r(x)