# Questions tagged [quintic-equations]

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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126 views

### Topological proof for the unsolvability of the quintic

Sorry, but in order to ask the question, you will have to view this video http://drorbn.net/dbnvp/AKT-140314.php. Here a topological proof for the unsolvability of the quintic is given, based on ideas ...
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### Loops through zero in Arnold's proof of the insolvability of the quintic

I have a question regarding Arnold's proof of the insolvability of the quintic (https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf). Here are my questions (but I'll give more context below)...
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### Find quintic roots of $x^ 5 = 1$

How many quintic roots, i.e. fifth roots, does the number 1 have? $$x^ 5 = 1$$ What are their values and how can one find them?
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### Proving equation has solution for every $c ≥ 0$

Task: Proof that the equation $x^5 − x = c$ has a solution for every $c \ge 0$ in the interval $[0, \infty)$. No idea where to start, anyone have any suggestions? Kind regards Anthony
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### Why couldn't Euler extend his method of solving a quartic to solve a quintic polynomial?

Euler creates a structure/identity (which are now generalised and known as Newtons identities) to solve a depressed quartic equation , where the coefficients are essentialy that of a cubic . We know ...
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### Why does transcendentals appear in the proof for Insolvability of Qunitic?

I'm currently studying Galois Theory using Fraleigh's abstract algebra. when proving the insolvability of the quintic, the book takes five independent transcendental elements over $\mathbb{Q}$ and ...
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### How to factorize the quintic polynomial $2x^5 + 6x^4 + 7x^3 + 21x^2 + 5x + 15$? [closed]

The polynomial is $$2x^5 + 6x^4 + 7x^3 + 21x^2 + 5x + 15$$ I want to find out the easiest way I can do factorize. Please show me the steps.
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### How to solve a quintic polynomial using elliptic functions with Mathematica?

I followed the exact steps from this forum post to solve quintic polynomials of the form: $x^5 - x + d$ But I got a different answer in number form from Mathematica than the original quintic ...
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### Is it possible to solve this reduced quintic equation?

Since there is no general formula for quintic equations, is there for a formula for quintic equations of the type: $$ax^5+bx^3+cx+d=0$$ or is it at least possible to solve: $$16x^5-20x^3+5x-q=0$$ in ...
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### quintic residue formula

let $\zeta_5$ a $5$-th primitive root of unity, and $\pi$ a prime in $\mathbb{Z}[\zeta_5]$ above $p$, I need to simplify the quintic residue formula: $\zeta_5^i(1+\zeta_5)^j \equiv X^5 (mod \pi$) with ...
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### Converting a root of a principal quintic to a root of the corresponding general quintic

I'm working on an implementation of a numeric solver for quintic equations and have a question regarding usage of the principal form. Say you have some quintic equation in general form. For example, ...
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### Where exactly Riemann surfaces and monodromy groups appear in this video?

This is regarding the proof of the unsolvability of the quintic equation (Abel-Ruffini theorem) which is explained really well in this short video. I also understand that the original proof by ...
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### Arnold's proof for the insolvability of the quintic

I am trying to understand Arnold's proof for the insolvability of the quintic from the manuscript: https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf which is actually well written. ...
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### When is the Galois group of a quintic not $S_5$ if we use this particular method?

I think that I am misunderstanding something fundamental about the technique used to decide if higher order polynomials are solvable by radicals using Galois theory. If we have a cubic it's not to bad ...
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### How to solve for imaginary roots of a quartic that cannot be factored and has no real roots?

Given the equation $4x^5 + 3x^4 + 2x^3 + x^2 + x - 11 = 0$, how do I find all complex roots? Fundamental theorem of algebra states that since this is a 5th degree polynomial, there are $5$ roots. I ...
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### Is quintic equations have anything to do with their coefficients?

I was reading about quintic equations and this came up: In the early nineteenth century, Paolo Ruffini (1765–1822) and Niels Henrik Abel (1802–1829) proved that no such general formulas ...
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### Geometrical shape of the solutions of the quintic equation in 3d

I understand the geometric meaning of the solutions of the cubics and quartics in the plane using combinations of equilateral stars with 3 and 4 arms, respectively. I heard that quintic equations can ...
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### Quintic polynomial with three real roots

I want to get a quintic polynomial $f(X) \in \mathbb{Q}[X]$ whose Galois group $\mathrm{Gal}(L/\mathbb{Q}) \cong S_5$ where $L$ is the splitting field of $f(X)$. One of strategies to get it is ...
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### What do I need to learn, to understand Galois' solution of the quintic?

I studied maths to degree level many years ago, but never got a "gut feel" for any of the "pure maths" side, because the way it was taught didn't fit at all with how I learn. I focused on numerical ...
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### Factor $x^5-x+15$

It's possible to factor $x^5-x+15$. WolframAlpha gives the answer of: $$(x^2+x+3)(x^3-x^2-2x+5)$$ According to the wikipedia article on quintic functions, the general form $x^5-x+a$ is factorable ...
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### Equation with Galois group twisted $S_{3}$

I note that a Galois group is not just a Galois group. Let $r_{1}$, $r_{2}$, $r_{3}$, $r_{4}$, denote the roots of a quartic equation. Then $x^4-5x^2+6$ has Galois group $Z_{2}^2$, where the ...
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### Number of real roots of a quintic equation

Find the number of connected components of the set $$\left\{x\in \mathbb R : x^3\left(x^2+5x-\frac{65}{3}\right)>70x^2-300x-297\right\}$$ under the usual topology on $\mathbb R$. We solve the ...
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### Condition for Quintic Reciprocity

Let $p=x^2+11x-1=1\pmod 5$ be a prime. Show that $x$ is a quintic residue $\pmod p$. It holds for $x<200$ and should hold for all such $x$. Any proof ideas? Thanks in advance.
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### Quintic reciprocity conjecture

Let $p=x^4 + 25x^2 + 125$ be a prime. Prove that $2$ is a quintic residue $\pmod p$, and therefore $y^5=2\pmod p$ is solvable. A similar example was first conjectured by Euler: If $p=x^2 + 27$ is a ...
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### Show: If some $x_i \neq x_1$ occurs as $\sigma(x_1)$ for a $\sigma \in$ Gal$(E:F)$ then each $x_i\neq x_1$ occurs as $\sigma(x_1)$

I'm currently studying Galois Theory (or am trying to do so), and since almost two weeks I try over and over again to solve a particular question from a textbook (J. Stillwell, Elements of Algebra, p. ...
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### Is $x^5-10x^3+20x= 8.58368$ solvable?

This quintic has 5 real roots, how do we find out if it is solvable and ,in that case, how to solve it? Is there a generally valid numeric approach?
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I'd seen it on wikipedia that no analytical solutions for quintic equation. However, I would like to ask are there some methods to solve it if we just consider the odd power terms shown as below: $ax^... 5answers 137 views ### Confirmation of Proof:$f(x)= x^5 - x - 16$has at most three real roots. I've been struggling to find a way to solve this question for a while now. I can prove that there is at least one real root using IVT but have no idea how to prove that there can exist more. ... 2answers 179 views ### Find all real solutions of the equation:$x^5+5x^3+5x+2017=0$I joined the math contest$1$week ago ( In Azerbaijan). There were$6$questions. Unfortunately, I could solve$1$question correctly.I know I can not write all the questions here, because It is ... 0answers 75 views ### Is the$t^5+15t+12=0$equation congruent with the Galois group? Is the quintic equation:$t^5+15t+12=0$congruent with the Galois group? And is it true that Lagrange tells us that there are other methods, which that we do not know for the quintic equation? 2answers 473 views ### Why is there no solution with radicals for the quintic equation$x^5-x+1=0$? [closed] I'm looking for solution for this equation $$x^5-x+1=0.$$ I know that there is no solution with radicals. But, I can not find possible solutions (in MSE or internet resource). I know Abel-... 1answer 48 views ### Finding the sum of the squares of the roots of a quintic I would like to solve the following problem.$a, b, c, d, e$are the roots of$P(x) = x^5 − 160x − 128 = 0$. Compute$a^5+b^5+c^5+d^5+e^5$. The first thing that came to mind was expand$(a+b+c+d+e)...
What are the steps used to find horizontal tangents of a function f(x) as defined: $$f(x)=(x+1)^2 (2x-3)^3$$ I know we have to calculate the derivative of the function, but how do we do that? (...