Homework 7

• Let C denote the set of complex numbers, with the usual addition (+) and multiplication (×) operations. Which of the following are true? If false, indicate what property does not hold.
  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}. Show that this set, with the usual addition and multiplication, is a field.
• Show that √ 3  is not an element of Q(√ 2 ).