Questions tagged [quintics]

Questions about polynomials with degree $5$. There is no general algebraic solution to these equations as proven by the Abel-Ruffini theorem, although some quintics are solvable.

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Is this 1803 result by Paolo Ruffini in relation to the solving a general quintic valid, and what assumptions might be made in his proof of it?

I am reading Paolo Ruffini's response in 1803 (see here) to a letter in 1802 from Pietro Abbati to Ruffini (see here), Abbati's letter suggesting improvements to Ruffini's lengthy 1799 'General Theory ...
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Formulae for polynomial equations of unsolvable degree by extension beyond radicals

There, of course, exists the linear formula and quadratic formula, and then there is the cubic formula of dubious usefulness, and then the quartic formula which is confined to the realm of mere ...
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Derivative and antiderivative of the Bring radical [duplicate]

The Bring radical of a real number $x$ is the unique real number $z=\operatorname{Br}(x)$ such that $z^5+z+x=0$. (Some I've seen say it's actually multivalued, with the real root for a real variable ...
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Question on Arnold's proof of the Abel-Ruffini Theorem

I am reading Arnold's proof from this paper by Leo Goldmakher. I think I have understood how the existence of single radicals in the cubic formula is ruled out in section 4, however, I don't ...
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Help needed with understanding Arnold's proof of Abel-Ruffini Theorem

I am trying to understand Arnold's proof with this document and this video. In section 5 the existence of the permutation (1 2 3 4 5) in the set of commutator loops is used to conclude that any ...
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Explicit example of a quintic threefold with 2875 distinct lines

A generic quintic threefold has 2875 lines. Is there an example of a quintic threefold explicitly defined by a polynomial that can be proven to have exactly 2875 distinct lines? I am particularly ...
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Finding Quintic Formula using any possible set of functions

Let's say that we are able to create a finite set of functions. The functions can be either single-argument ones (for example Cosine) or two-argument ones (for example Sum). The fuctions can only ...
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Simplest unsolvable quintic with one real root

I am aware that $t^5-t-1$ is unsolvable, but the proof I have seen involves a theorem linking its Galois group with the Galois group of its reduction mod $p$. If I wish to have a simpler proof (that ...
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5th degree equation roots relation

Let $a,b,c,d,e$ be the roots of $x^5+x^4+2x^3+4x^2+7x+1=0$ the question is:find the value of $$\frac{1}{2-a}+\frac{1}{2-b}+\frac{1}{2-c}+\frac{1}{2-d}+\frac{1}{2-e}$$ my try is in below with respect ...
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Can a quintic equation be solved without needing a "piece of paper the size of a large asteroid"(!)?

I remember from a long time ago reading a paper regarding the solution of quintic polynomial equations using hypergeometric functions. In particular, the methods are based around the solution of the ...
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What would the field corresponding to a Galois group of $S_5$ look like? [closed]

I'm wondering what other tools there are aside from radicals can be used to extend fields in the context of solving polynomials. Since $S_5$ isn't solvable, constructing a field with a Galois group of ...
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1 vote
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Are general degree 5 polynomials solvable only with only elementary functions and the Lambert W function?

I know it is not solvable in terms of radicals, and I don't know if it is known if it can be solved with only elementary functions of any kind. But can it be solved using only elementary functions and ...
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Is there a proof showing super roots and super logarithms won't lead to a solution for the quintic?

So I am learning about tetrations and I just learned that tetrations are not elementary functions. When I heard that I remembered back to the statement that there is no general solution to the quintic ...
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A better way to find the radius of the complex roots of $(z + 1)^5 = 32z^5$?

I was gnawing on this problem today: All the complex roots of $(z + 1)^5 = 32z^5,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle. I solved this by first dividing ...
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It's well known that it is impossible to solve a generic quintic equation in terms of radicals involving its coefficients. However: what's the "most generic" quintic equation that is still ...
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If a real number can be expressed in terms of complex solutions of quintics, can it be expressed in terms of real solutions of quintics?

I'm considering a notion of quintic-constructible numbers. This is a follow-up from the cubic case. Suppose a real number $\alpha$ is contained in a tower of fields \alpha\in\mathbb F_m\supset\...
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Finding $k$ such that $a$ is a real root $x^5-x^3+x-2=0$ with $[a^6]=3k$. ($[x]$ denotes the Greatest Integer function.)

If $a$ is a real root of the equation $x^5-x^3+x-2=0$ such that $[a^6]=3k$. Find $k$. Here, $[x]$ denotes the Greatest Integer function. My try: First I plugged in $1$ in the equation. I got $-1$, ...
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Can a Quintic Polynomial Be Solved?

I am well aware of the fact that a general quintic polynomial cannot be solved in radicals. However, is there a known way to obtain a formula with other functions (e.g. infinitely nested radicals)?
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