Let be a finite dimensional vector space over and a linear transformation. Suppose that and for some prime number Prove that

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- Jul 2nd 2009, 10:41 AMNonCommAlgAlgebra, Problems For Fun (29)
Let be a finite dimensional vector space over and a linear transformation. Suppose that and for some prime number Prove that

- Jul 2nd 2009, 10:54 PMBruno J.
Let .

Expressing in its Jordan Form we find that has zero trace.

Moreover since is a th root of 1, the diagonal entries of must all be of the form .

Finally it is an easy exercise to show that if then ; hence there are as many of each of the elements on the main diagonal of .

Setting , this implies the characteristic polynomial of is , using the identity . Hence . - Jul 3rd 2009, 09:50 AMBruno J.
Proof of the above "easy exercise"

We have

Hence if then . But are linearly independent over hence for all . - Jul 3rd 2009, 05:22 PMNonCommAlg