An equation
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 be contain under the root sign one or more of the previous radicals,
![{\displaystyle {\begin{matrix}{\begin{cases}\xi _{1}={\sqrt[{p_{1}}]{r_{1}}},\\\xi _{2}={\sqrt[{p_{2}}]{r_{2}(\xi _{1})}},\\\xi _{3}={\sqrt[{p_{3}}]{r_{3}(\xi _{1},\,\xi _{2})}},\\\cdots \qquad \cdots \\\xi _{m}={\sqrt[{p_{m}}]{r_{m}(\xi _{1},\,\xi _{2},\,\ldots ,\,\xi _{m-1})}},\end{cases}}\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a553df83ba40bf82ff628eb9a2a6be19b020a0e)
where generally\, 
\, is an element of the field
\, but no 
'th power of an element of this field.\, Because of the formula ![{\displaystyle {\sqrt[{jk}]{r}}={\sqrt[{j}]{\sqrt[{k}]{r}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/665debd6aa0f89b9e69d435aedc2c1022b4de799)
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
![{\displaystyle \xi _{1}={\sqrt {\left({\frac {q}{2}}\right)^{2}\!+\!\left({\frac {p}{3}}\right)^{3}}},\quad \xi _{2}={\sqrt[{3}]{-{\frac {q}{2}}\!+\!\xi _{1}}},\quad \xi _{3}={\sqrt {-3}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f289bf93354395453499bf0c27418babb66cede5)
In fact, as we consider also the equation (4), the roots may be expressed as
