Conjugate element (field theory)

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In mathematics, in particular field theory, the conjugate elements or algebraic conjugates of an algebraic element α, over a field extension L/K, are the roots of the minimal polynomial p_K,α(x) of α over K. Conjugate elements are commonly called conjugates in contexts where this is not ambiguous. Normally α itself is included in the set of conjugates of α.

Equivalently, the conjugates of α are the images of α under the field automorphisms of L that leave fixed the elements of K. The equivalence of the two definitions is one of the starting points of Galois theory.

The concept generalizes the complex conjugation, since the algebraic conjugates over ${\displaystyle \mathbb {R} }$ of a complex number are the number itself and its complex conjugate.