Answer to Question #238418 in Abstract Algebra for 123

Question #238418

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


1
Expert's answer
2021-09-21T11:30:34-0400

Solution:

We know that in the field of quotients [(a,b)]={(c,d):ad=bc}, [(0,1)][(a, b)]=\{(c,d):ad=bc\},\ [(0,1)] is the additive neutral element and [(a,b)]+[(c,d)]=[(ad+bc,bd)].[(a,b)]+[(c,d)]=[(ad+bc,bd)]. Taking into account that 01=b20,0\cdot 1=b^2\cdot 0, and hence [(0,b2)]=[(0,1)],[(0,b^2)]=[(0,1)], we conclude that [(a,b)]+[(a,b)]=[(ab+ba,b2)]=[(ab+ab,b2)]=[(0,b2)]=[(0,1)],[(−a, b)]+[(a, b)]=[(-ab+ba,b^2)]=[(-ab+ab,b^2)]=[(0,b^2)]=[(0,1)], and therefore [(a,b)][(−a, b)] is the additive inverse for [(a,b)][(a, b)] in the field of quotients.


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