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algebraically solvable equation (Definition)

An equation

$\displaystyle x^n+a_1x^{n-1}+\ldots+a_n = 0,$ (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   $K(\xi_1,\,\xi_2,\,\ldots,\,\xi_m)$  which is obtained of $K$ by adjoining to it in succession certain suitable radicals $\xi_1,\,\xi_2,\,\ldots,\,\xi_m$.  Each radical may be contain under the root sign one or more of the previous radicals,
\begin{align*}\begin{cases} \xi_1 = \sqrt[p_1]{r_1},\ \xi_2 = \sqrt[p_2]{r_2(\... ...m = \sqrt[p_m]{r_m(\xi_1,\,\xi_2,\,\ldots,\,\xi_{m-1})}, \end{cases}\end{align*}    
where generally  $r_k(\xi_1,\,\xi_2,\,\ldots,\,\xi_{k-1})$  is an element of the field $K(\xi_1,\,\xi_2,\,\ldots,\,\xi_{k-1})$  but no $p_k$'th power of an element of this field.  Because of the formula

$\displaystyle \sqrt[jk]{r} = \sqrt[j]{\sqrt[k]{r}}$
one can, without hurting the generality, suppose that the indices $p_1,\,p_2,\,\ldots,\,p_m$ are prime numbers.

Example.  Cardano's formulae show that all roots of the cubic equation $y^3+py+q = 0$ are in the algebraic number field which is obtained by adjoining to the field  $\mathbb{Q}(p,\,q)$  successively the radicals

$\displaystyle \xi_1 = \sqrt{\left(\frac{q}{2}\right)^2\!+\!\left(\frac{p}{3}\right)^3}, \quad \xi_2 = \sqrt[3]{-\frac{q}{2}\!+\!\xi_1}, \quad \xi_3 = \sqrt{-3}.$
In fact, as we consider also the equation (4), the roots may be expressed as
\begin{align*}\begin{cases} \displaystyle y_1 = \xi_2-\frac{p}{3\xi_2}\ \displ... ...\cdot\xi_2-\frac{-1\!+\!\xi_3}{2}\cdot\!\frac{p}{3\xi_2} \end{cases}\end{align*}    



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Cross-references: algebraic, formula, power, operations, field

This is version 2 of algebraically solvable equation, born on 2009-04-18, modified 2009-04-18.
Object id is 662, canonical name is AlgebraicallySolvableEquationsDefinition.
Accessed 236 times total.

Classification:
Physics Classification: 02.10.De (Algebraic structures and number theory)
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