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 cube roots of the number one are: The latter two roots are conjugate elements in Q[i√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 pK,α, containing α.
If L is any normal extension of K containing α, then by definition it already contains such a splitting field.
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.
There are quantitative forms of this, stating more precisely bounds (depending on degree) on the largest absolute value of a conjugate that imply that an algebraic integer is a root of unity.