दिया है,
`omega ``= "cos" (2pi)/(3) + i" sin " (2pi)/(3)`
`omega ``= "cos" (pi - (pi)/(3)) + i" sin " (pi - (pi)/(3)) `
`omega ``= "-cos" ((pi)/(3)) + i" sin " ((pi)/(3)) `
`omega ``=-1/2 + isqrt3/2`
`omega ``=(-1 + isqrt3)/2`
यह मान इकाई का घनमूल है, इकाई के घनमूल के 3 मान हम जानते है
`(-1 + isqrt3)/2`, `(-1 - isqrt3)/2`, `1`
इसे निम्न प्रकार से लिख सकते है
`omega`, `omega^2`, `1`
जहाँ `1+``omega``+omega^2``=0` तथा `omega^3 = 1`
निम्न समीकरण के लिए `z` के मानो की संख्या
`|(z+1,omega,omega^(2)),(omega,z+ omega^(2),1),(omega^(2),1,z+ omega)| =0`
`C_1 -> ``C_1 + C_2 + C_3`
`|(z+1+omega+omega^2,omega,omega^(2)),(z+1+omega+omega^2,z+ omega^(2),1),(z+1+omega+omega^2,1,z+ omega)| =0`
`|(z,omega,omega^(2)),(z,z+ omega^(2),1),(z,1,z+ omega)| =0`
`R_2->R_2 - R_1` व `R_3->R_3 - R_1`
`|(z,omega,omega^(2)),(0,z+ omega^(2)-omega,1-omega^(2)),(0,1-omega,z+ omega-omega^(2))| =0`
`C_1` के सापेक्ष सारणिक का विस्तार करने पर
`zxx[(z+ omega^(2)-omega)xx(z+ omega-omega^(2))-(1-omega^(2))xx(1-omega)]``-0+0` `=0`
`zxx[zxxz ``+ zxxomega -zxxomega^(2) ``+ omega^(2)xxz ``+ omega^(2)xxomega ``-omega^(2)xxomega^(2) ``-omegaxxz ``- omegaxxomega ``+ omegaxxomega^(2) ``- (1-omega ``- omega^(2) ``+ omega^3)]``=0`
`zxx[z^2 ``+ zxxomega -zxxomega^(2) ``+ zxxomega^(2) ``+ omega^(3) ``-omega^(3)xxomega ``-zxxomega ``- omega^2 ``+ omega^(3) ``- 1+ omega ``+ omega^(2) ``- omega^3]``=0`
`zxx[z^2 ``+ cancel(zxxomega) -cancel(zxxomega^(2)) ``+ cancel(zxxomega^(2)) ``+ 1 ``- cancel(omega) ``-cancel(zxxomega) ``- cancel(omega^2) ``+ 1 ``- 1+ cancel(omega) ``+ cancel(omega^(2)) ``- 1]``=0`
`z^3 = 0`
`z = 0`
अतः यहाँ `z` का एक ही मान प्राप्त होगा।
