Conjugate element (field theory)

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 $\mathbb {R}$ of a complex number are the number itself and its complex conjugate.

Example

The cube roots of the number one are:

{\sqrt[{3}]{1}}={\begin{cases}1\\[3pt]-{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i\\[5pt]-{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}i\end{cases}}

The latter two roots are conjugate elements in $Q [i \sqrt 3]$ with minimal polynomial

\left(x+{\frac {1}{2}}\right)^{2}+{\frac {3}{4}}=x^{2}+x+1.

Properties

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.

In summary, the conjugate elements of α are found, in any normal extension L of K that contains K(α), as the set of elements g(α) for g in Aut(L/K). The number of repeats in that list of each element is the separable degree [L:K(α)]_sep.

A theorem of Kronecker states that if α is a nonzero algebraic integer such that α and all of its conjugates in the complex numbers have absolute value at most 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.

References

David S. Dummit, Richard M. Foote, Abstract algebra, 3rd ed., Wiley, 2004.

External links

Weisstein, Eric W. "Conjugate Elements". MathWorld.