Conjugate element (field theory)
{{#invoke:Hatnote|hatnote}}Template:Main other {{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} 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
- pK,α(x)
of α over K.
Example
The cube roots of the number one are:
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63986d451190db4ab0c72f77c59b00f8d0c6a900)
The latter two roots are conjugate elements in L/K = Q[√3, i]/Q[√3] with minimal polynomial
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 pK,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 FTemplate:' that maps polynomial p to pTemplate:' can be extended to an isomorphism of the splitting fields of p over F and pTemplate:' over FTemplate:', 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 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.
References
- David S. Dummit, Richard M. Foote, Abstract algebra, 3rd ed., Wiley, 2004.