with coefficients in a field , is algebraically solvable, if some of its roots may be expressed with the elements of by using rational operations (addition, subtraction, multiplication, division) and root extractions. I.e., a root of (1) is in a field which is obtained of by adjoining to it in succession certain suitable radicals . Each radical may contain under the root sign one or more of the previous radicals,
where generally is an element of the field but no ’th power of an element of this field. Because of the formula
one can, without hurting the generality, suppose that the indices are prime numbers.
Example. Cardano’s formulae show that all roots of the cubic equation are in the algebraic number field which is obtained by adjoining to the field 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).