Conjugate element (field theory)

In mathematics, in particular field theory, the conjugate elements 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 also called Galois conjugates, or simply conjugates. Normally α itself is included in the set of conjugates of α.

The cube roots of the number one are:

The latter two roots are conjugate elements in L/K = Q[√3, i]/Q[√3] with minimal polynomial

If K is given inside an algebraically closed field C, then the conjugates can be taken inside C. If no such C is specified, one can take the conjugates in some relatively small field L. The smallest possible choice for L is to take a splitting field over K of p_K,α, containing α. If L is any normal extension of K containing α, then by definition it already contains such a splitting field.

Given then a normal extension L of K, with automorphism group Aut(L/K) = G, and containing α, any element g(α) for g in G will be a conjugate of α, since the automorphism g sends roots of p to roots of p. Conversely any conjugate β of α is of this form: in other words, G acts transitively on the conjugates. This follows as K(α) is K-isomorphic to K(β) by irreducibility of the minimal polynomial, and any isomorphism of fields F and F' that maps polynomial p to p' can be extended to an isomorphism of the splitting fields of p over F and p' over F', respectively.

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