Sunday, 23 October 2016

abstract algebra - What is wrong with this proof of Wedderburn's little theorem?



Wedderburn's little theorem every finite domain A is a field.




Proof Let x be a nonzero element of A. Because A is finite, there
exist positive integers n, k such that xn=xn+k. It is easy to
see by induction that the set E={xi:iN}
does not contain 0; it follows therefore from xn(1xk)=0
that xk=1. Thus, xk1 is the inverse of x (when k=1, x has
inverse 1).



All the proofs I have seen of this result are much more sophisticated than mine. Hence, I am doubting its correctness and could use a second opinion.


Answer




Wedderburn's little theorem states as you wrote above, that a finite domain is a field. A field is a commutative domain A, such that every nonzero xA has a multiplicative inverse. Your proof only shows that any finite domain is a skew field. You must also prove that A is commutative, which needs more sophisticated arguments.


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