algebraically solvable

An equation

xn+a1xn-1++an=0, (1)

with coefficients aj in a field K, is algebraically solvable, if some of its roots ( may be expressed with the elements of K by using rational operations (addition, subtractionPlanetmathPlanetmath, multiplication, division) and root extractions.  I.e., a root of (1) is in a field  K(ξ1,ξ2,,ξm)  which is obtained of K by adjoining ( to it in succession certain suitable radicalsMathworldPlanetmathPlanetmath ξ1,ξ2,,ξm.  Each radical may under the root sign one or more of the previous radicals,

{ξ1=r1p1,ξ2=r2(ξ1)p2,ξ3=r3(ξ1,ξ2)p3,  ξm=rm(ξ1,ξ2,,ξm-1)pm,

where generally  rk(ξ1,ξ2,,ξk-1)  is an element of the field K(ξ1,ξ2,,ξk-1)  but no pk’th power of an element of this field.  Because of the formula


one can, without hurting the generality, suppose that the indices ( p1,p2,,pm are prime numbersMathworldPlanetmath.

Example.  Cardano’s formulae show that all roots of the cubic equationy3+py+q=0  are in the algebraic number fieldMathworldPlanetmath which is obtained by adjoining to the field  (p,q)  successively the radicals


In fact, as we consider also the equation (4), the roots may be expressed as



  • 1 K. Väisälä: Lukuteorian ja korkeamman algebran alkeet.  Tiedekirjasto No. 17.   Kustannusosakeyhtiö Otava, Helsinki (1950).
Title algebraically solvable
Canonical name AlgebraicallySolvable
Date of creation 2015-04-15 13:48:08
Last modified on 2015-04-15 13:48:08
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Definition
Classification msc 12F10
Synonym algebraic solvability
Synonym solvable algebraically
Related topic RadicalExtension
Related topic KalleVaisala