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BoolQ / can a polynomial have a square root of a variable

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can a polynomial have a square root of a variable
passage · full text (1,741 characters)
There may be several meanings of ``solving an equation''. One may want to express the solutions as explicit numbers; for example, the unique solution of 2x -- 1 = 0 is 1/2. Unfortunately, this is, in general, impossible for equations of degree greater than one, and, since the ancient times, mathematicians have searched to express the solutions as algebraic expression; for example the golden ratio ( 1 + 5 ) / 2 (\displaystyle (1+(\sqrt (5)))/2) is the unique positive solution of x 2 − x − 1 = 0. (\displaystyle x^(2)-x-1=0.) In the ancient times, they succeeded only for degrees one and two. For quadratic equations, the quadratic formula provides such expressions of the solutions. Since the 16th century, similar formulas (using cube roots in addition to square roots), but much more complicated are known for equations of degree three and four (see cubic equation and quartic equation). But formulas for degree 5 and higher eluded researchers for several centuries. In 1824, Niels Henrik Abel proved the striking result that there are equations of degree 5 whose solutions cannot be expressed by a (finite) formula, involving only arithmetic operations and radicals (see Abel--Ruffini theorem). In 1830, Évariste Galois proved that most equations of degree higher than four cannot be solved by radicals, and showed that for each equation, one may decide whether it is solvable by radicals, and, if it is, solve it. This result marked the start of Galois theory and group theory, two important branches of modern algebra. Galois himself noted that the computations implied by his method were impracticable. Nevertheless, formulas for solvable equations of degrees 5 and 6 have been published (see quintic function and sextic equation).

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task_id
f020dc4b-d53c-576a-9cff-02a5c2a80538
task_key
default--validation--f020dc4b-d53c-576a-9cff-02a5c2a80538
task_revision_id
1
config / split
default / validation

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