
theorem Th32: :: DimCenter:
  for R being finite Skew-Field holds 0 < dim VectSp_over_center R
proof
  let R be finite Skew-Field;
  set q = card the carrier of center R;
  set n = dim VectSp_over_center R;
  set Rs = MultGroup R;
  card R = q |^ n by Th31;
  then
A1: card Rs = (q |^ n) - 1 by UNIROOTS:18;
  now
    assume
A2: n = 0;
    q |^ n = q #Z n by PREPOWER:36;
    then card Rs = 1 - 1 by A1,A2,PREPOWER:34;
    hence contradiction by GROUP_1:45;
  end;
  hence thesis;
end;
