Hi,

problem:

Let $\displaystyle F$ be the set of all (ordered) pairs $\displaystyle (\alpha,\beta)$ of real numbers.

(a) If addition and multiplication are defined by

$\displaystyle (\alpha,\beta)+(\gamma,\delta)=(\alpha+\gamma,\bet a+\delta)$

and

$\displaystyle (\alpha,\beta)(\gamma,\delta)=(\alpha\gamma,\beta\ delta)$,

does $\displaystyle F$ become a field?

(b) If addition and multiplication are defined by

$\displaystyle (\alpha,\beta)+(\gamma,\delta)=(\alpha+\gamma,\bet a+\delta)$

and

$\displaystyle (\alpha,\beta)(\gamma,\delta)=(\alpha\gamma-\beta\delta,\alpha\delta+\beta\gamma)$,

is $\displaystyle F$ a field then?

(c) What happens (in both the preceding cases) if we consider ordered pairs of complex numbers instead?

attempt (at (a)):

I have gone through all the axioms for a field and am unable to find one that fails.

For example, the multiplicative inverse:

$\displaystyle (\alpha,\beta)\frac{1}{\alpha,\beta} \Rightarrow (\alpha,\beta)\left(\frac{1}{\alpha},\frac{1}{\bet a}\right)=(1,1)$

One of my friends told me that this is not a field, but I fail to see why.

Thanks!