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algebraically solvable
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(Definition)
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An equation
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(1) |
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,
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. V¨AISÄLÄ: Lukuteorian ja korkeamman algebran alkeet. Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).
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"algebraically solvable" is owned by pahio.
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(view preamble)
Cross-references: algebraic number field, cubic equation, Cardano's formulae, prime numbers, power, radicals, root, division, multiplication, subtraction, addition, operations, rational, field, coefficients, equation
There are 3 references to this entry.
This is version 4 of algebraically solvable, born on 2008-03-03, modified 2008-03-04.
Object id is 10361, canonical name is AlgebraicallySolvable.
Accessed 541 times total.
Classification:
| AMS MSC: | 12F10 (Field theory and polynomials :: Field extensions :: Separable extensions, Galois theory) |
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Pending Errata and Addenda
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![\begin{align*}\begin{cases}\xi_1 = \sqrt[p_1]{r_1},\\ \xi_2 = \sqrt[p_2]{r_2(\xi... ...m = \sqrt[p_m]{r_m(\xi_1,\,\xi_2,\,\ldots,\,\xi_{m-1})}, \end{cases}\end{align*}](http://images.planetmath.org:8080/cache/objects/10361/l2h/img8.png "\begin{align*}\begin{cases}\xi_1 = \sqrt[p_1]{r_1},\\ \xi_2 = \sqrt[p_2]{r_2(\xi... ...m = \sqrt[p_m]{r_m(\xi_1,\,\xi_2,\,\ldots,\,\xi_{m-1})}, \end{cases}\end{align*}")
![$\displaystyle \sqrt[jk]{r} = \sqrt[j]{\sqrt[k]{r}}$](http://images.planetmath.org:8080/cache/objects/10361/l2h/img12.png "$\displaystyle \sqrt[jk]{r} = \sqrt[j]{\sqrt[k]{r}}$")
![$\displaystyle \xi_1 = \sqrt{\left(\frac{q}{2}\right)^2\!+\!\left(\frac{p}{3}\right)^3}, \quad \xi_2 = \sqrt[3]{-\frac{q}{2}\!+\!\xi_1}, \quad \xi_3 = \sqrt{-3}.$](http://images.planetmath.org:8080/cache/objects/10361/l2h/img16.png "$\displaystyle \xi_1 = \sqrt{\left(\frac{q}{2}\right)^2\!+\!\left(\frac{p}{3}\right)^3}, \quad \xi_2 = \sqrt[3]{-\frac{q}{2}\!+\!\xi_1}, \quad \xi_3 = \sqrt{-3}.$")
