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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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.
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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 ...
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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. ...
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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?
1
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2answers
144 views

Are there some methods to solve the quintic equation?

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^...
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5answers
101 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. ...
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2answers
161 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 ...
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0answers
65 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?
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2answers
284 views

Solution for $x^5-x+1=0$ [closed]

I'm looking for solution for this equation: $$x^5-x+1=0$$ I know There are not solution with radicals. But, I can not find possible solution. (In MSE and internet resource) And Can you show me why ...
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1answer
45 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)...
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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? (...
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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 ...
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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 ...