finte fields in maxima

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   Date: Sat, 5 Jan 2013 09:21:04 +0800
   From: Jianrong Li <lijr07 at gmail.com>
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   Dear Leo,

   Thank you very much for your help. The field F which contains a copy of the
   field of order 2^n for each n \geq 1 is the algebraic closure of the
   2-element field F_2. Let a be a matrix over F_2.  Is there some function
   which can determine a is diagonalizable over F or not?

   For example, let
   a=
   [[0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0],
   [1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0],
   [1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
   [1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
   [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
   [1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0],
   [0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0],
   [1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0],
   [0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
   [0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0],
   [1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0],
   [0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0],
   [1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0],
   [1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0],
   [1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0],
   [0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0]];

   Does "a is diagonalizable over F" mean "the 0, 1, in a are considered as
   elements in F and there are invertible matrix P over F such that PaP^{-1}
   is diagonal? Do we have 1+1=0 in F?

   Thank you very much.
   With best wishes,
   Jianrong.

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