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⋅1=b2⋅0, and hence [(0,b2)]=[(0,1)], we conclude that [(−a,b)]+[(a,b)]=[(−ab+ba,b2)]=[(−ab+ab,b2)]=[(0,b2)]=[(0,1)], and therefore [(−a,b)] is the additive inverse for [(a,b)] in the field of quotients.
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