Do sets of all pairs of real numbers form fields?
Let be the set of all (ordered) pairs of real numbers.
(a) If addition and multiplication are defined by
does become a field?
(b) If addition and multiplication are defined by
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.