# Resolvent

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A resolvent of an algebraic equation of degree is an algebraic equation , with coefficients that rationally depend on the coefficients of , such that, if the roots of this equation are known, the roots of the given equation can be found by solving simpler equations of degrees not exceeding . A rational expression itself is sometimes called a resolvent.

Let be a separable polynomial over a field with Galois group , and let be a normal subgroup of . Let be a rational expression in that remains invariant under all permutations of the roots belonging to , and let . Then is a root of some equation with coefficients from , the Galois group of which is a proper quotient group of . Thus, solving the equation reduces to solving the equation and solving the equation over the field .

For example, in order to solve an equation of degree : (every equation of degree is reducible to this form), the following cubic resolvent is used: Its roots are related to the roots by the relations , , . The roots are determined by the Cardano formula, which also makes it possible to determine .

Successive application of the resolvent method permits one to solve any equation with a solvable Galois group by reduction to solving a chain of equations with cyclic Galois groups. Lagrange's resolvent is used in solving the latter.

Let be an equation over a field with a cyclic Galois group of order , and let contain a primitive -th root of unity . For an element which belongs to the splitting field of the polynomial (cf. Splitting field of a polynomial), and for a character from into the group of -th roots of unity, Lagrange's resolvent is defined by the formula: (*)

Let be one of the roots of the polynomial and let run through the characters of . Then for the system of linear equations (*) the roots can be determined if the Lagrange resolvents are known for all characters of .

For the relation is fulfilled, showing that the elements and , for any integer , are invariant under and are therefore uniquely defined rational expressions in the coefficients of the polynomial and the root . If generates the group of characters of , then the following equalities hold: and for .

Any algebraic equation that is irreducible over a given field (see Galois theory) and that is such that as a result of the adjunction of one of its roots to this field a field is obtained that contains all roots of the equation , is called a Galois resolvent of .

How to Cite This Entry:
Resolvent. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Resolvent&oldid=12785
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article