Resolvent

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

• The resolvent operator in operator theory: $R(z;A)= (A-zI)^{-1}.\,$
• The resolvent set in operator theory, the set of points where an operator is "well-behaved".
• The resolvent (Galois theory) for a permutation group G, which is a polynomial whose coefficients depend polynomially of the coefficients of a given polynomial p and has a rational root if and only if the Galois group of p is included in G. The resolvents have been introduced by Joseph Louis Lagrange and systematically used by Évariste Galois. Nowadays there are yet a fundamental tool to compute Galois groups. The simplest examples of resolvents are
  • X² − Δ where Δ is the discriminant, which is a resolvent for the alternating group.
  • The cubic resolvent of a quartic equation which is a resolvent for the dihedral group of 8 elements.