suppose and then obviously is not invertible. thus which gives us: so i.e. has no non-zero nilpotent element.

hence where are finite fields. if we're done. so suppose that since is odd, must be odd for all consider two cases:

case 1: for some without loss of generality we assume that choose and let then is clearly a non-invertible element

of but because contradiction!

case 2: if then which is impossible because the problem assumes that therefore now let then is a

non-invertible element of and which is again a contradiction!Q.E.D.