algebraically solvable equation
An equation
 |
(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
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