Abstract Algebra Theory and Applications - Computer Science ...

16.1 FIELDS OF FRACTIONS 279and[a, b] · [c, d] = [ac, bd],respectively. The next lemma demonstrates that these operations are independentof the choice of representatives from each equivalence class.Lemma 16.2 The operations of addition and multiplication on F D are welldefined.Proof. We will prove that the operation of addition is well-defined. Theproof that multiplication is well-defined is left as an exercise. Let [a 1 , b 1 ] =[a 2 , b 2 ] and [c 1 , d 1 ] = [c 2 , d 2 ]. We must show thator, equivalently, that[a 1 d 1 + b 1 c 1 , b 1 d 1 ] = [a 2 d 2 + b 2 c 2 , b 2 d 2 ](a 1 d 1 + b 1 c 1 )(b 2 d 2 ) = (b 1 d 1 )(a 2 d 2 + b 2 c 2 ).Since [a 1 , b 1 ] = [a 2 , b 2 ] and [c 1 , d 1 ] = [c 2 , d 2 ], we know that a 1 b 2 = b 1 a 2 andc 1 d 2 = d 1 c 2 . Therefore,(a 1 d 1 + b 1 c 1 )(b 2 d 2 ) = a 1 d 1 b 2 d 2 + b 1 c 1 b 2 d 2= a 1 b 2 d 1 d 2 + b 1 b 2 c 1 d 2= b 1 a 2 d 1 d 2 + b 1 b 2 d 1 c 2= (b 1 d 1 )(a 2 d 2 + b 2 c 2 ).Lemma 16.3 The set of equivalence classes of S, F D , under the equivalencerelation ∼, together with the operations of addition and multiplicationdefined byis a field.[a, b] + [c, d] = [ad + bc, bd][a, b] · [c, d] = [ac, bd],Proof. The additive and multiplicative identities are [0, 1] and [1, 1], respectively.To show that [0, 1] is the additive identity, observe that[a, b] + [0, 1] = [a1 + b0, b1] = [a, b].□