Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs
Theorem
Let $\map f z = a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0$ be a polynomial over complex numbers where $a_0, \ldots, a_n$ are real numbers.
Let $\alpha \in \C$ be a root of $f$.
Then $\overline \alpha$ is also a root of $f$, where $\overline \alpha$ denotes the complex conjugate of $\alpha$.
That is, all complex roots of $f$ appear as conjugate pairs.
Proof 1
Let $\alpha \in \C$ be a root of $f$.
Then $\map f \alpha = 0$ by definition.
Suppose $\alpha$ is wholly real.
Then by Complex Number equals Conjugate iff Wholly Real:
- $\alpha = \overline \alpha$
and so $\overline \alpha$ is a root of $f$.
Now let $\alpha \in \C$ not be wholly real.
By definition of complex conjugate, we have that:
- $\overline 0 = 0$
and so:
- $\map f \alpha = \overline {\map f \alpha}$
From Conjugate of Polynomial is Polynomial of Conjugate:
- $\overline {\map f \alpha} = \map f {\overline \alpha}$
from which it follows that:
- $\map f {\overline \alpha} = 0$
That is, $\overline \alpha$ is also a root of $f$.
$\blacksquare$
Proof 2
Let $\alpha = p + q i$.
Let $p + q i$ be expressed in exponential form as $\alpha = r e^{i \theta}$.
As $\alpha = r e^{i \theta}$ satisfies $\map f \alpha = 0$, it follows that:
- $a_n r^n e^{n i \theta} + a_{n - 1} r^{n - 1} e^{\paren {n - 1} i \theta} + \dotsb + a_1 r e^{i \theta} + a_0 = 0$
Taking the conjugate of both sides:
- $a_n r^n e^{-n i \theta} + a_{n - 1} r^{n - 1} e^{-\paren {n - 1} i \theta} + \dotsb + a_1 r e^{-i \theta} + a_0 = 0$
it follows that $\overline \alpha = p - q i$ is also a root of $f$.
If any of the $a_k$ are complex, then the conjugate of $f$ is not the same polynomial as $f$.
Therefore the result holds only for real coefficients.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): conjugate (of a complex number)