algebraic conjugates

Let $L$ be an algebraic extension of a field $K$, and let ${\alpha }_1\in L$ be algebraic over $K$. Then ${\alpha }_1$ is the root of a minimal polynomial $f(x)\in K[x]$. Denote the other roots of $f(x)$ in $L$ by ${\alpha }_2$, ${\alpha }_3,\mathrm{\dots },{\alpha }_n$. These (along with ${\alpha }_1$ itself) are the algebraic conjugates of ${\alpha }_1$ and any two are said to be algebraically conjugate.

The notion of algebraic conjugacy is a special case of group conjugacy in the case where the group in question is the Galois group of the above minimal polynomial, viewed as acting on the roots of said polynomial.

Title	algebraic conjugates
Canonical name	AlgebraicConjugates
Date of creation	2013-03-22 13:58:39
Last modified on	2013-03-22 13:58:39
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	9
Author	mathcam (2727)
Entry type	Definition
Classification	msc 11R04
Synonym	algebraically conjugate
Synonym	conjugate
Related topic	ComplexConjugate
Related topic	ConjugationMnemonic