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It is not generally possible to determine the roots of a polynomial whose grade is bigger than 4 in terms of roots and basic operations. But I heard that it is possible to give a criteria whether a polynomial has such a solution.

For instance Wikipedia tells that the roots of a polynomial of degree $5$ are expressible in terms of roots and basic operations, if it is representable in the form

$$x^5 + \frac{5\mu^4(4\nu + 3)}{\nu^2 + 1}x + \frac{4\mu^5(2\nu + 1)(4\nu + 3)}{\nu^2 + 1} = 0,$$

where $\mu$ and $\nu$ are rational numbers.

Given the case that the polynomials roots are expressible in such a form, is it possible to give an algorithm that computes the solution in this form?

I am not an expert on that topic, just a student who is interested in maths.

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    Are you asking the same thing that was asked in http://math.stackexchange.com/questions/33612/how-to-solve-polynomials ? You may want to have a look at that question...2011-05-18

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Yes, it's called Galois Theory.

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    I read the article and think, that you can certainly show that a polynomial's roots are describable in terms of radicands using the Galois Theory, but it doesn't explains you how to find them.2011-05-18
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    @FUZxxl, see for instance http://math.stackexchange.com/questions/27877/galois-groups-of-polynomials-and-explicit-equations-for-the-roots2011-05-18
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While one cannot express the terms of polynomials of degree 5 and higher with basic operations and roots (Abel-Ruffini), it is possible to explicitly calculate the roots of polynomials of degree five and higher if you use other tools.

See this MO question.

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    My question was, if the polynomial *is* expressable in terms of radicals, is it possible to give an algorithm to find them?2011-05-19
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Yes, assuming an expression for a root exists, there is a simple algorithm that will find it: define an ordering on all such expressions (e.g. a lexical ordering), and test each one until you find a root of the input polynomial.

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    On the other hand, the problem of deciding whether a particular expression is a root of the given polynomial may not be so easy. I presume that it can be done numerically using some super-exponential error bound in terms of the length of the expression resulting from substituting the candidate root into the polynomial. Or is it guaranteed that it will evaluate to 0 symbolically? I'm really not sure.2011-06-02
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    I'm pretty well convinced now that a symbolic method will work for the test, essentially because of algebraic independence (e.g. the cube root of p will never equal the square root of q). If anyone knows otherwise, I welcome a correction.2011-06-02
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    If you can find the minimal polynomial of the expression you're testing, you can evaluate the polynomial in terms of that. For example, for a polynomial $P(x)$ and and expression whose minimal polynomial is $x^2-2$, let $a$ represent a number such that $a^2-2=0$ and evaluate $P(a)$, replacing every factor of $a^2$ with 2. All the terms if this evaluation will cancel iff the expression is a root of $P$.2018-05-30
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    I believe I have a method for finding the minimal polynomial of any radical expression, which is mostly symbolic, augmented with a little numeric computation (which can be done in a way that doesn't cast any doubt on the answer). There may be a way to eliminate the numerical part, but I haven't found it.2018-05-30
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    (The minimal polynomial formulation is called for because it turns out that when working with radical expressions directly. it's not at all straightforward to find the necessary simplifications, if they exist, when the expression is complicated. Try taking a radical expression for a primitive fifth root of unity to the fifth power and you might see what I mean.)2018-05-30