How to prove that if a quartic equation ( with real coefficients ) has 4 imaginary roots they all will be in conjugate pairs?
I proved this fact for qudratic equation in the following way , let a qudratic equation have a imaginary root (q is not 0), and let other root be . Now here sum of roots will be a real number lets say R, . So finally the roots are, , hence proved. I tried to prove this fact for quartic equation in a similar way but could not reach to the conclusion. Please guide me by answering by my method or by suggesting any other simple method to prove that if a quartic equation has all imaginary roots then they will occur in conjugate pairs.