LetΦ be the ring homomorphism from Z[x] to Z given by Φ(f(x)) =f(1). Find a polynomial g(x) in Z[x] such that Ker Φ =<g(x)>. Isthere more than one possibility for g(x)? To what familiar ring isZ[x]/Ker Φ isomorphic? Do this exercise with Z replaced by Q.

If f: Z→ Z is the map defined by f(x)=2x, Is f a ring homomorphism when Z has its usual ring operations? How would you prove that? If not, what could be a counter-example?

(c) Let f(x) and g(x) be nonzero polynomials in R[x], of degrees m and n, respectively. Prove that deg(f(x) g(x)) = m + n. (d) Give a counterexample to the multiplicative inverse law for the ring R[x] of polynomials in x with real coefficients. Explain why your counterexample works.

Define a ring homomorphism 4: R[x] → RX R, (ƒ(x)) = (fƒ(0), ƒ(1)) In other words, y maps f(x) to its evaluations at x = and at x = = 1. (a) Describe the kernel of y, in terms of polynomials having certain roots/zeroes. (b) Find polynomials f(x) and g(x) in R[x] such that f(0) = 1, f(1) = 0, and g(0) = 0, g(1) = 1 (Note: There are many different options!) Let a, b € R, and define h(x) = af (x) + bg(x). Show that y(h(x)) = (a,b). (c) Using part (b), explain why is surjective. (d) What does the 1st Isomorphism Theorem say, when applied to ?