# 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.

51 questions
Filter by
Sorted by
Tagged with
1answer
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 ...
0answers
64 views

### 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)...
3answers
53 views

### 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?
2answers
23 views

### 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
0answers
60 views

### 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 ...
2answers
149 views

0answers
81 views

0answers
67 views

### 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 ...
1answer
140 views

### 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.
1answer
44 views

### 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 ...
0answers
60 views

### 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 ...
0answers
39 views

### 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 ...
0answers
60 views

### 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, ...
0answers
24 views

### 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 ...
1answer
153 views

### 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. ...
0answers
75 views

### 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 ...
0answers
54 views

### 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 ...
1answer
41 views

### 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 ...
0answers
54 views

### 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 ...
1answer
98 views

### 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 ...
1answer
74 views

### 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 ...
4answers
283 views

### 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 ...
1answer
96 views

### 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 ...
1answer
143 views

### 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 ...
1answer
135 views

### 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.
2answers
222 views

### 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 ...
1answer
173 views

### 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. ...
2answers
59 views

### 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?
2answers
191 views

2answers
346 views

### Determine the Horizontal tangents of f(x)

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? (...
1answer
641 views

### Explanation of the Tschirnhausen transformation

I am studying the resolution of the quintic equations, which involves the so-called Tschirnhausen transform. The idea is to cancel the fourth and third degree coefficients by a change of variable of ...