Conjugate element (field theory)

"Conjugate elements" redirects here. For conjugate group elements, see Conjugacy class.

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

Equivalently, the conjugate elements of α are the images of α by the K-automorphisms of L/K.

Example

The cube roots of the number one are:

${\sqrt[{3}]{1}}={\begin{cases}\ \ 1\\[8pt]-{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i\\[8pt]-{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}i\end{cases}}$

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

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.

If L/K is separable, then the product of the conjugates of α is the norm of α, and belongs to K. More generally, the symmetric functions of the conjugates of α are the coefficients of the minimal polynomial of α, and hence belong to K.

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.

Conjugates of a polynomial

Assume that f(x) is a separable irreducible polynomial in K[X], and that there exists an extension M/K and a polynomial g in M[X] such that g divides f in M[X]. If L denotes the splitting field of f over K, then L/K is Galois, and L[X]/K[X] is isomorphic to L/K. In particular, the polynomial g is algebraic over K[X], and hence has conjugate elements over K[X]. Actually, the set of conjugates of g is

$\{\sigma (g):\ \sigma \in {\hbox{Gal}}(L/K)\}$.

Properties

It is natural to believe that the product of the conjugates of g is equal to f; nevertheless, this is false in general, unless g is irreducible and f is primitive (that is, L/K is spanned by a single root of f).

In general, the product of the conjugates of g is equal to cfⁿ, where c belongs to the constant field K and n is a natural number.).

References

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

External links

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