PlanetMath: algebraic conjugates

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algebraic conjugates

(Definition)

Let be an algebraic extension of a field , and let $\alpha_1\in L$ be algebraic over . Then $\alpha_1$ is the root of a minimal polynomial $f(x)\in K[x]$ . Denote the other roots of in by $\alpha_2$ , $\alpha_3, \ldots, \alpha_n$ . 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.

"algebraic conjugates" is owned by mathcam.

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Other names:

algebraically conjugate, conjugate

Attachments:: conjugate fields (Definition) by pahio

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Cross-references: polynomial, Galois group, group, conjugacy, minimal polynomial, root, algebraic, field, algebraic extension
There are 17 references to this entry.

This is version 6 of algebraic conjugates, born on 2003-10-02, modified 2006-07-24.
Object id is 4749, canonical name is AlgebraicConjugates.
Accessed 4358 times total.

Classification:

AMS MSC:

11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers)

Pending Errata and Addenda

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