1.20 The set of ordered pair of fields whose addition defined by (a, b) + (c, d) = (a+c, b+d) and multiplication defined by (a, b)(c, d) = (ac, bd) is not a field

Proof:
The identity element of the set is (1, 1) since (a, b)(1, 1) = (a, b).
The inverse of (1, 0)(x, y) = (x, 0) = (1, 1) which implies 0 = 1 but this is a contradiction.