## Field if I define The plane

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?
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 Quote by Bachelier 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?
No, you have zero divisors.

 would the multiplication as described here make ℂ a field?
How would you do i2 = -1?

 Quote by pwsnafu No, you have zero divisors.
You mean like (0,7).(8,0)

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## Field if I define The plane

Yes, neither (7, 0) nor (0, 8) is the additive identity but neither has a mulitplicative inverse.
 thanks. Basically ℂ will fail to be an integral domain in the first place under this operation.