Hi,

problem:

Let be the set of all (ordered) pairs of real numbers.

(a) If addition and multiplication are defined by

and

,

does become a field?

(b) If addition and multiplication are defined by

and

,

is 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:

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

Thanks!