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| In ], a '''resolvent''' for a permutation group ''G'' is a ] whose coefficients depend polynomially on the coefficients of a given polynomial ''p'' and has, roughly speaking, a rational root if and only if the ] of ''p'' is included in ''G''. More exactly, if the Galois group is included in ''G'', then the resolvent has a rational root, and the converse is true if the rational root is a ]. | In ], a discipline within the field of ], a '''resolvent''' for a ] ''G'' is a ] whose coefficients depend polynomially on the coefficients of a given polynomial ''p'' and has, roughly speaking, a ] root if and only if the ] of ''p'' is included in ''G''. More exactly, if the Galois group is included in ''G'', then the resolvent has a rational root, and the converse is true if the rational root is a ]. | ||
| Resolvents were introduced by ] and systematically used by ]. Nowadays they are yet a fundamental tool to compute ]s. The simplest examples of resolvents are | Resolvents were introduced by ] and systematically used by ]. Nowadays they are yet a fundamental tool to compute ]s. The simplest examples of resolvents are | ||
| * <math>X^2-\Delta</math> where <math>\Delta</math> is the ], which is a resolvent for the ]. In the case of a ] this resolvent is sometimes called '''resolvent quadratic'''; its roots appear explicitly in the formulas for the roots of a cubic equation. | * <math>X^2-\Delta</math> where <math>\Delta</math> is the ], which is a resolvent for the ]. In the case of a ] this resolvent is sometimes called '''resolvent quadratic'''; its roots appear explicitly in the formulas for the roots of a cubic equation. | ||
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| For every equation the roots may be expressed in terms of radicals and of a root of a resolvent for a resoluble group, because, the Galois group of the equation over the field generated by this root is resoluble. | For every equation the roots may be expressed in terms of radicals and of a root of a resolvent for a resoluble group, because, the Galois group of the equation over the field generated by this root is resoluble. | ||
| == References == | == References == | ||
| * {{cite book |title=Algebraic Theories |first= Leonard E.|last=Dickson |authorlink=Leonard Eugene Dickson |publisher= Dover Publications Inc|location= New York|year= 1959|isbn=0-486-49573-6 |pages= |page= ix+276|}} | * {{cite book |title=Algebraic Theories |first= Leonard E.|last=Dickson |authorlink=Leonard Eugene Dickson |publisher= Dover Publications Inc|location= New York|year= 1959|isbn=0-486-49573-6 |pages= |page= ix+276|}} | ||
Revision as of 18:23, 8 November 2012
In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a rational root if and only if the Galois group of p is included in G. More exactly, if the Galois group is included in G, then the resolvent has a rational root, and the converse is true if the rational root is a simple root. Resolvents were introduced by Joseph Louis Lagrange and systematically used by Évariste Galois. Nowadays they are yet a fundamental tool to compute Galois groups. The simplest examples of resolvents are
- where is the discriminant, which is a resolvent for the alternating group. In the case of a cubic equation this resolvent is sometimes called resolvent quadratic; its roots appear explicitly in the formulas for the roots of a cubic equation.
- The cubic resolvent of a quartic equation which is a resolvent for the dihedral group of 8 elements.
- The Cayley resolvent is a resolvent for the maximal resoluble Galois group in degree five. It is a polynomial of degree 6.
where 
is the These three resolvents have the property of being always separable, which means that, if they have a multiple root, then the polynomial p is not irreducible. It is not known if there is an always separable resolvent for every group of permutations.
For every equation the roots may be expressed in terms of radicals and of a root of a resolvent for a resoluble group, because, the Galois group of the equation over the field generated by this root is resoluble.
References
- Dickson, Leonard E. (1959). Algebraic Theories. New York: Dover Publications Inc. p. ix+276. ISBN 0-486-49573-6.
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