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The Collaborative International Dictionary
Resolvent

Resolvent \Re*solv"ent\ (-ent), a. Having power to resolve; causing solution; solvent.

Resolvent

Resolvent \Re*solv"ent\, n. [L. resolvens, p. pr. of resolvere: cf. F. r['e]solvant. See Resolve.]

  1. That which has the power of resolving, or causing solution; a solvent.

  2. (Med.) That which has power to disperse inflammatory or other tumors; a discutient; anything which aids the absorption of effused products.
    --Coxe.

  3. (Math.) An equation upon whose solution the solution of a given pproblem depends.

Wiktionary
resolvent

a. Able to resolve (separate) the constituents of a mixture n. 1 Any substance or material able to resolve the constituents of a mixture; a solvent. 2 (context medicine English) That which has power to disperse inflammatory or other tumours; a discutient; anything which aids the absorption of effused products. 3 (context mathematics English) An equation upon whose solution the solution of a given problem depends.

WordNet
resolvent

n. a liquid substance capable of dissolving other substances; "the solvent does not change its state in forming a solution" [syn: solvent, dissolvent, dissolver, dissolving agent]

Wikipedia
Resolvent

In mathematics, resolvent meaning "that which resolves" may refer to:

  • Resolvent formalism in operator theory
  • Resolvent set in operator theory, the set of points where an operator is "well-behaved"
  • Feller process#Resolvent in probability theory
  • Resolvent (Galois theory) of an equation for a permutation group, in particular:
    • Resolvent quadratic of a cubic equation
    • Resolvent cubic of a quartic equation

In logic:

  • Resolvent (logic), the clause produced by a resolution
  • In the consensus theorem, the term produced by a consensus in Boolean logic
Resolvent (Galois theory)

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 still a fundamental tool to compute Galois groups. The simplest examples of resolvents are

  • X − Δ where Δ is the discriminant, which is a resolvent for the alternating group. In the case of a cubic equation, this resolvent is sometimes called the quadratic resolvent; 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.

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.