Homework 8

• Algebraic structure of the set of polynomials.
  Let R[x] denote the set of all polynomials with real coefficients, with the usual addition and multiplication operations. Determine which of the following structures this set has. If it has a certain structure, specify the identity and inverse elements for the above operations; do not write out proofs of properties such as commutativity, asscociativity, and distributivity (these are lengthy but very straightforward). If it does not have a certain structure, identify at least one property that does not hold.
  • Is R[x] a group under addition? If so, is it an abelian group?
  • Is it a group under multiplication?
  • Is it a ring under addition and multiplication? If so, is it a commutative ring?
  • Is it a field under addition and multiplication?
  • One number set (N, Z, Q, R, or C) has exactly the same algebraic structures as R[x]. Which one?
  • Does there exist a bijective function between R[x] and the number set that you identified in the previous question, that preserves both operations (addition and multiplication)? If so, construct it. If not, explain why not.