On May 23, 2009, Edwin Woollett wrote
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" Here is code for exploration of the case n = 4.
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(%i9) pr : factor( rectform(s) );
(%o9) [(%i-1)/sqrt(2),-(%i+1)/sqrt(2),-(%i-1)/sqrt(2),(%i+1)/sqrt(2)]
(%i10) ratsimp( pr^4 );
(%o10) [-1,-1,-1,-1]
(%i11) ratsimp( pr - p[1] );
(%o11) [0,0,0,0] "
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Additional steps in exploration: multiply the
list pr by ( - 1), %i, or ( - %i ) and show
you get the same four complex numbers,
just in a different order.
(%i12) pr1 : -pr;
(%o12) [-(%i-1)/sqrt(2),(%i+1)/sqrt(2),(%i-1)/sqrt(2),-(%i+1)/sqrt(2)]
(%i13) ratsimp( pr1^4 );
(%o13) [-1,-1,-1,-1]
(%i14) pr2 : ratsimp(-%i*pr);
(%o14) [(%i+1)/sqrt(2),(%i-1)/sqrt(2),-(%i+1)/sqrt(2),-(%i-1)/sqrt(2)]
(%i15) ratsimp( sort(pr2) - sort(pr1));
(%o15) [0,0,0,0]
(%i16) ratsimp( pr2^4 );
(%o16) [-1,-1,-1,-1]
(%i17) pr3 : ratsimp(%i*pr);
(%o17) [-(%i+1)/sqrt(2),-(%i-1)/sqrt(2),(%i+1)/sqrt(2),(%i-1)/sqrt(2)]
(%i18) ratsimp(pr3^4);
(%o18) [-1,-1,-1,-1]
(%i19) ratsimp( sort(pr3) - sort(pr1));
(%o19) [0,0,0,0]
Ted Woollett