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Theorem 2.1. Every finite integral domain is a field., , Proof: The proof is based on the fact that since R is an integral domain it has a cancellation, law. Having a cancellation law means that, , ab=ac => b=c (7), , To see why that any finite integral domain R is a field, consider R = {r,r?,r°,...,r”}, where r* #0 forl<k<n. Since R is finite we will have rk — r! for some k and | such, that k > 1. Then, , rk = gl # Ris a finite integral domain, => rerklaperhl # factor out r, => pela, # use cancellation law (cancel r, Equation 7), => rept 2a. el? # factor out r, => pho? = pl-2 # use cancellation law (cancel r, Equation 7), : # iterate 1 — 1 times, => rH al #..., => repktip. 7 # factor out r, => peta, # use cancellation law (cancel r, Equation 7), => petiy # ro =1, , So r*—! = 1, If k—1 =1 then r a unit since r*—! = r! = 1 so r7} is 1, Otherwise k-1 > 1, , and ré-! = 1 => rk“ = h, So r-} = r*-! and every r #0 € R has an inverse. Thus, every non-zero r € R is a unit and so R is a field.