Answer to Question #238418 in Abstract Algebra for 123

Question

Prove that [(-a,b)] is the additive inverse for [(a,b)] in the field of quotients. NOTE: these are equivalence classes

Answer 1

Solution:

We know that in the field of quotients $[(a, b)]=\{(c,d):ad=bc\},\ [(0,1)]$ is the additive neutral element and $[(a,b)]+[(c,d)]=[(ad+bc,bd)].$ Taking into account that $0\cdot 1=b^2\cdot 0,$ and hence $[(0,b^2)]=[(0,1)],$ we conclude that $[(−a, b)]+[(a, b)]=[(-ab+ba,b^2)]=[(-ab+ab,b^2)]=[(0,b^2)]=[(0,1)],$ and therefore $[(−a, b)]$ is the additive inverse for $[(a, b)]$ in the field of quotients.