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Field if I define The plane

  1. Jan 30, 2013 #1
    if I define The plane: ##F = ℝ## x ##ℝ = \{ (a, b) | a, b ∈ ℝ \} ##

    and define addition and multiplication as:
    (a, b) + (c, d) := (a + c, b + d)
    (a, b) · (c, d) := (ac, bd)

    Then ##F## is a field. right?

    would the multiplication as described here make ℂ a field?
     
  2. jcsd
  3. Jan 30, 2013 #2

    pwsnafu

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    Re: Field

    No, you have zero divisors.

    How would you do i2 = -1?
     
  4. Jan 31, 2013 #3
    Re: Field

    You mean like (0,7).(8,0)
     
  5. Jan 31, 2013 #4

    HallsofIvy

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    Yes, neither (7, 0) nor (0, 8) is the additive identity but neither has a mulitplicative inverse.
     
  6. Jan 31, 2013 #5
    thanks. Basically ℂ will fail to be an integral domain in the first place under this operation.
     
    Last edited: Jan 31, 2013
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