An equation

xn + a 1xn−1 + … + a n = 0,

(1)

with coefficients a_j 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, subtraction, multiplication, division) and root extractions. I.e., a root of (1) is in a field K(ξ₁, ξ₂, …, ξ_m) which is obtained of K by adjoining to it in succession certain suitable radicals ξ₁, ξ₂, …, ξ_m. Each radical may be contain under the root sign one or more of the previous radicals,

where generally r_k(ξ₁, ξ₂, …, ξ_k−1) is an element of the field K(ξ₁, ξ₂, …, ξ_k−1) but no p_k’th power of an element of this field. Because of the formula

one can, without hurting the generality, suppose that the indices p₁, p₂, …, p_m are prime numbers.

Example. Cardano’s formulae show that all roots of the cubic equation y³ + py + q = 0 are in the algebraic number field which is obtained by adjoining to the field ℚ(p, q) successively the radicals

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