Conjugate element (field theory) - Wikipedia

Abstraction: Roots of minimal polynomial as algebraic conjugates over a field

Key points:

• The algebraic conjugates of an element α over a field extension L/K are the roots of the minimal polynomial p_{K,α}(x) of α over K.
• Equivalently (when L/K is normal), conjugates are the images of α under all field automorphisms of L fixing K — a foundational idea in Galois theory.
• The Galois group Aut(L/K) acts transitively on the conjugates of α; the number of distinct conjugates equals [K(α):K] for separable extensions.
• Generalizes complex conjugation: the complex conjugates of a complex number are the two roots of its minimal polynomial over R.
• Kronecker's theorem: if a nonzero algebraic integer and all its complex conjugates have absolute value ≤ 1, then it is a root of unity.

Connections: Leopold Kronecker · Conjugate Elements · Galois Theory · Field Extension · Minimal Polynomial

Source: http://en.wikipedia.org/wiki/Conjugate_(algebra)