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I have $s\neq 0, r \in \mathbb{R}$ I also have $(a,b)\cdot(c,d)=(ac-bd(r^{2}+s^{2}),ad+bc+2rbd)$

Question, does this satisfy the multiplication requirements to be considered a field.

I tried using (1,0)x(0,1) Since this gives (0,1), shouldn't this count as a field since the inverse isn't an instance of division by 0?

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How is addition defined - pointwise? Also, \times=$\times$ and u\cdot v=$u\cdot v$. –  anon Nov 30 '12 at 4:18
    
To address your last two lines, @MathScratch: you first must find an element candidate to be the multiplicative unit. –  DonAntonio Nov 30 '12 at 4:21
    
Don, multiplicative unit for which part? The ordered pair (0,1)? –  MathScratch Nov 30 '12 at 4:29
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@MathScratch : multiplicative unit with respect to the operation you've proposed. (Exercise: show that $e=(1,0)$ is indeed that unit: you should be able to easily show that $e\cdot(a,b) = (a,b)$ for every element $(a,b)$.) –  Steven Stadnicki Nov 30 '12 at 6:07

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