# Questions tagged [quintic-equations]

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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 ...
0answers
45 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
33 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
39 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
50 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
65 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
233 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
81 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
103 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
96 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
176 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
155 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
55 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
144 views

2answers
289 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
354 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 ...
2answers
128 views

### Computation of the ultraradical

The ultraradical of a real number, also called the Bring radical, is the unique real solution of the quintic equation $$y=x^5+x.$$ This function is used in the resolution of general quintic ...