If <math>p</math> is a [[polynomial]] with [[real number|real]] coefficients, and <math>p(z) = 0</math>, then <math>p(\overline{z}) = 0</math> as well. Thus non-real roots of real polynomials occur in complex conjugate pairs. (See the [[complex conjugate root theorem]] article.)
TheAz functionkomplex számokból komplex számokba képező <math>\phif(z) = \overline{z}</math> from <math>\mathbb{C}</math> to <math>\mathbb{C}</math> is[[függvény]] [[continuousfolytonos functionfüggvény|continuousfolytonos]]. EvenNoha thoughigen itegyszerű, appears to be a "tame"nem [[well-behavedanalitikus függvény|analitikus]] function, itmert is not [[holomorphic]]; it reverses orientation whereas holomorphic functions locally preserve orientation. It is [[bijective]] and compatible with the arithmetical operations, and hence is a [[field (mathematics)|field]] [[automorphism]]. As it keeps the real numbers fixed, it is an element of the [[Galois group]] of the [[field extension]] <math>\mathbb{C}/\mathbb{R}</math>. This Galois group has only two elements: <math>\phi</math> and the identity on <math>\mathbb{C}</math>. Thus the only two field automorphisms ofÍgy <math>\mathbb{C}</math>-nek thatpontosan leavekét theolyan realautomorfizmusa numbersvan, fixedami area thevalósokat identityfixen maphagyja: andaz complexidentitás conjugationés a konjugálás.
