Compute a quotient in a given extension of QQ

Let \omega be a primitive cube root of 1. (That is, \omega^3 = 1, but no smaller power of \omega is 1.) Find a,b,c \in \mathbb{Q} such that 2 - \omega = (a+b\omega + c\omega^2)(3 + \omega^2).


Carrying out the multiplication on the right hand side and noting that \omega^3 = 1, we get the following system of equations: a+3c = 0, 3b+c = -1, and 3a+b = 2. Solving this system yields a = 3/4, b = -1/4, and c = -1/4. Indeed, we can easily verify that (\omega^2+3)(\frac{3}{4} - \frac{1}{4}\omega - \frac{1}{4}\omega^2) = 2 - \omega.

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