with coefficients in a field , is algebraically solvable, if some of its roots (http://planetmath.org/Equation) 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 (http://planetmath.org/FieldAdjunction) to it in succession certain suitable radicals . Each radical may 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 (http://planetmath.org/Root) are prime numbers.
In fact, as we consider also the equation (4), the roots may be expressed as
- 1 K. Väisälä: Lukuteorian ja korkeamman algebran alkeet. Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).
|Date of creation||2015-04-15 13:48:08|
|Last modified on||2015-04-15 13:48:08|
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