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algebraically solvable
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(Definition)
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An equation
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(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,
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 $$\sqrt[jk]{r} = \sqrt[j]{\sqrt[k]{r}}$$ 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 $$\xi_1 = \sqrt{\left(\frac{q}{2}\right)^2\!+\!\left(\frac{p}{3}\right)^3}, \quad \xi_2 = \sqrt[3]{-\frac{q}{2}\!+\!\xi_1}, \quad \xi_3 = \sqrt{-3}.$$ In fact, as we consider also the equation (4), the roots may be expressed as
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- 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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Cross-references: algebraic number field, cubic equation, Cardano's formulae, prime numbers, formula, 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 1447 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/js/img2.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*}")
