Conjugate element (field theory)

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A conjugate root is a root of a polynomial function that has a conjugate which is also a root of the polynomial function. If P is a polynomial and a + bi is a root of p, then a - bi is also a root of P.

For example, the cube roots of the number one are:

The latter two roots are conjugate pairs.

In mathematics, in particular field theory, the conjugate elements of an algebraic element α, over a field K, are the (other) roots of the minimal polynomial

P_K,α(t)

of α over K. If K is given inside an algebraically closed field C, then the conjugates can be taken inside C. Usually one includes α itself in the set of conjugates.

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. Assuming only that P is a separable polynomial, this means that we can take L to be a Galois extension.

Given then a Galois extension L of K, with Galois group 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 of α is of this form: in other words, G acts transitively on the conjugates. This follows because it is true for the Galois group of the splitting field, and G maps surjectively to that group by basic properties of the Galois correspondence.

In summary, assuming separability, the conjugate elements of α are found, in any finite Galois extension L of K that contains K(α), as the set of elements g(α) for g in Gal(L/K). The number of repeats in that list of each element is the degree [L:K(α)] which is also [L:K]/d where d is the degree of P.

A theorem of Kronecker states that if α is an algebraic integer such that α and all of its conjugates in the complex numbers have absolute value 1, then α is a root of unity. 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.