Homework 9

• 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 required 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?
  • 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?
  • Which number set (N, Z, Q, R, or C) has exactly the same algebraic structures as R[x]?