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prove that a finite integral domain is also a feild(MAths-Ring&Field)
Administrator
on 2010-05-18 09:30:00A finite integral domain is a field. Proof: Let R be a finite integral domain. Let a be nonzero element of R . Define a function :RR by (r)=ar . Suppose (r)=(s) for some rsR . Then ar=as , which implies a(r−s)=0 . Since a=0 and R is a cancellation ring, we have r−s=0 . So r=s , and hence is injective. Since R is finite and is injective, by the pigeonhole principle we see that is also surjective. Thus there exists some bR such that (b)=ab=1R , and thus a is a unit. Thus R is a finite division ring. Since it is commutative, it is also a field.
AVISHEK
on 2010-05-19 09:30:00also solved in das pal..