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Field if I define The plane 
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#1
Jan3013, 10:14 PM

P: 376

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
Jan3013, 11:54 PM

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#3
Jan3113, 12:21 AM

P: 376




#4
Jan3113, 09:46 AM

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PF Gold
P: 39,510

Field if I define The plane
Yes, neither (7, 0) nor (0, 8) is the additive identity but neither has a mulitplicative inverse.



#5
Jan3113, 11:35 AM

P: 376

thanks. Basically ℂ will fail to be an integral domain in the first place under this operation.



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