1.1 The Field Properties 5- The set {O, 1, 2, 3} with the operations + and · defined by the tables:
+^0 1 2 3 0 1 2 3
0 0 1 2 3 0 0 0 0 0
1 1 2 3 0 1 0 1 2 3
2 2 3 0 1 2 0 2 0 2
3 3 0 1 2 3 0 3 2 1- (Project) Consider the set C = { (a, b) : a, b E JR} of "complex num-
bers,'' where we define
(a, b) + ( c, d) = (a + c, b + d)
(a, b)(c, d) = (ac - bd, ad+ be),using the usual addition and multiplication of elements of R(a) Prove that the set C is a field.
(b) Prove that the set {(a, 0): a E JR} is a subfield of C.
(c) In what sense is this subfield essentially "the same" as R [Having
done P art (b) you know that the identity element of C is 1 = (1, 0).
Continue, showing that \fa E JR, a and (a, 0) can be regarded as "the
same."]
(d) Let i = (0, 1) and prove that i^2 = -1, where we are identifying -1
with ( -1, 0) as justified in ( c) above.
(e) Explain how it is true that \fa, b E JR, the complex number (a, b) can
be written in the form a+ bi.CONSEQUENCES OF THE FIELD AXIOMS
We now proceed to show that the familiar algebraic rules governing addi-
tion, subtraction, multiplication, and division hold in any field.
Theorem 1.1.2 In any field F, the cancellation laws hold:
(a) ifx+y=x+z (ory+x=z+x), theny=z;
(b) ifxy=xz (oryx=zx) andxo;iO, theny=z.