Intereting Posts

Conditional probability for an exponential variable
Do we gain anything interesting if the stabilizer subgroup of a point is normal?
Disjoint Refinement
Cauchy product of two absolutely convergent series is absolutely convergent. (Rudin PMA Ch. 3 ex 13)
Another trigonometric proof…?
Proving that 1- and 2-d simple symmetric random walks return to the origin with probability 1
Proving or disproving that an inequality implies another inequality.
Bounding a solution of an ODE with a small source
Proving the Riemann Hypothesis and Impact on Cryptography
Conditional expectation on components of gaussian vector
Is the following scheme for generating $p_n=(1/3)^n$ stable or not. $p_n=(5/6)p_{n-1}-(1/6)p_{n-2}$.
Explain why $e^{i\pi} = -1$ to an $8^{th}$ grader?
Jacobian matrix rank and dimension of the image
Is there a difference between $y(x)$ and $f(x)$
Can $A, B$ fail to commute if $e^A=e^B=e^{A+B}=id$?

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

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

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- Explanation on arg min
- General McNugget problem
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- Finding all roots of polynomial system (numerically)
- How do Gap generate the elements in permutation groups?

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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- Probability Calculations on Highway

Yes, it’s called Galois Theory.

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.

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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