Homework 7

• Let C denote the set of complex numbers, with the usual addition (+) and multiplication (×) operations. Which of the following are true?
  For those that are true, you have to make sure for yourself that all the required properties hold, however, do not write out the proofs in your homework. Just state that they are true.
  For those that are false, indicate at least one property that does not hold. Think for yourself how you would prove it, but do not write out the proof in your homework.
  1. (C,+) is a group
  2. (C,×) is a group
  3. (C-{0},+) is a group
  4. (C-{0},×) is a group
  5. (C,+,×) is a ring
  6. (C,+,×) is a commutative ring with identity
  7. (C,+,×) is a field
• Let Q(√ 2 ) = { a + b*√ 2  | a and b are in Q}. Using the definition of a field, convince yourself that this set, with the usual addition and multiplication, is a field. Do not write out the proof in your homework.
• Prove that √ 3  is not an element of Q(√ 2 ).