\(\frac{a + b \omega + c \omega^2}{b + c \omega + a \omega^2 }\)
= \(\frac {\omega(a + b \omega + c \omega^2)}{\omega(b + c \omega + a \omega^2) }\)
Now, we simplify the numerator:
= ω(a + bω + cω2) = ωa + bω2 + cω3
Since ω3 = 1, we can replace ω3 with 1:
= ωa + bω2 + c.
Next, we simplify the denominator:
ω(b + cω + aω2) = ωb + cω2 + aω3.
Again, replacing ω3 with 1 gives us:
= ωb + cω2 + a.
= \(\frac{\omega a + b \omega^2 + c}{a + \omega b + c \omega^2}\)
ωa + bω2 + c = a + ωb + cω2.
\(\frac{a + b \omega + c \omega^2}{b + c \omega + a \omega^2 } = \omega\)
