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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Transforming the reduced sextic $x^6+x^2+ax+b$ into a quintic
The Bring-Jerrard quintic: $$x^5+x = a$$
For $a≠0$. After some change we get: $$\frac x a = a^{-4/5}\left(1- \frac x a\right)^{1/5}$$
Expanding the RHS using binomial theorem because $\frac x a = \...
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Calculating the discriminant of a quintic polynomial
I am trying to calculate the discriminant of the polynomial $f = X^5 + pX^2 + q \in K[X]$ where $K$ is a field. I define the discriminant as $\text{disc } f = \prod_{i < j} (\alpha_i - \alpha_j)^2$ ...
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Does it make sense to say a quintic equation has a "repeated quadratic root"?
Regarding the following question from a Further Maths textbook:
The equation $2x^5+x^4+36x^3+18x^2+162x+81=0$ has a repeated
quadratic root.
a) Show that $x=3i$ is a solution of the equation
b) Fully ...
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Is there a way to solve quintics of the form $ax^5+bx^4+c=0$?
It is known that if we only consider the first four decimal digits of the following (no rounding), we have that this holds:$$\pi^4+\pi^5=e^6\label1\tag1$$However, if we use even one more decimal place,...
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Understanding Abel Ruffini theorem
Due to efforts of Abel, Ruffini, we know that there does not exist a general formula for a quintic equation. But given a specific quintic, say $x^5+5x^2-97x+1001=0$, Can there exist a root for it in ...
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Quintic polynomial with positive discriminant
This is a sequel to that question where I learned why a real quintic polynomial with positive discriminant has either one or five distinct real roots.
As a followup, given such a polynomial (i.e. real,...
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Galois theory and Lagrange's method on quintics
I have just begun studying Galois Theory from Stewart's book and got some questions with some conclusions regarding Lagrange's methods when discussing the quintics. For reference, I will put below ...
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Relationship of the roots of two polynomials both which have the same constant and x term but vary in the x^3 term by a function of the x coefficient [closed]
Given that we know the roots of the first equation is there anyway to use them or the relationship between these two quintic polynomials to discern the roots of the second equation.
is there any ...
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The transformation relating two quintic polynomials
I apologise if this a silly question. Is there a transformation relating the following two graphs:
$x^5 + -3 \sqrt{\frac{e}{5}}x^3 + ex + f$
$x^5 + ex + f$
if so is there any way to derive it like ...
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Skolem’s “Theorem 9” and the associated equation
In Diophantine Equations (page 209), Mordell writes:
[Skolem] proved that the equation
$$x^5+2y^5+4z^5-10xy^3z+10x^2yz^2=1$$
has at most six solutions in integers $x,y,z$,
and of these three are ...
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On generalizing Ramanujan cosine sums to $\sqrt[5]{x_1}+\sqrt[5]{x_2}+\sqrt[5]{x_3}+\sqrt[5]{x_4}+\sqrt[5]{x_5} = \sqrt[5]{z}$?
I. Cubic
In 2013, I asked a question regarding an identity by Ramanujan of form,
$$\sqrt[3]{0+2\cos\tfrac{2\pi}{7}}+\sqrt[3]{0+2\cos\tfrac{4\pi}{7}}+\sqrt[3]{0+2\cos\tfrac{6\pi}{7}} = \sqrt[3]{+5-3\,\...
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Quintic equation with integer coefficients
I am looking for a way to find a closed form of the real root of the quintic eq. with integer coefficients:
$x^5+3x^4+4x^3+x-1=0$.
According to the numerical calculation the root $x_0\approx 0....
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On factoring the resolvent equations of solvable quintics, septics, etc?
I. p = 11
The minimal polynomial of $x=2\cos\frac{2\pi}{11}$ is,
$$x^5 +x^4 −4x^3 −3x^2 +3x+1 = 0$$
As is known, Method 1 solves it in the form,
$$x = \frac15\left(A+y_1^{1/5}+y_2^{1/5}+y_3^{1/5}+y_4^{...
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The solution to the principal quintic via the Brioschi and Rogers-Ramanujan cfrac $R(q)$
In this interesting post, it quotes a paper that to write down the complete solution (not in radicals) of the general quintic, one would need "a piece of paper as big as a large asteroid". ...
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Clarifying the meaning of insolvability of quintic
Insolvability of quintic says that no analogue of quadratic formula is available for quintic equations. For quadratic equations, the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ gives you all the ...
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How large is the gap in Ruffini's 1813 proof that there is no general quintic formula?
I'm reading Ruffini's final attempt at showing there is no general quintic formula which appeared in 1813 see here. (This is a much shorter proof than his first proof of 1799 in his Teoria generale ...
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Quintic polynomial with only positive roots
The Abel-Ruffini theorem states that the solutions to a general polynomial equation of the form$$x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0$$ have no algebraic expression in terms of the coefficients of ...
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Integrating $\int \frac{dx}{\sqrt{x^5+ax^3+bx^2+cx+d}}$.
$$\int \frac{dx}{\sqrt{x^5+ax^3+bx^2+cx+d}}$$
I know we have $E,K,$ and $\Pi$ to integrate quartics under the radical, but I wonder what functions quintics yield? Are there any known?
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Can De Moivre's quintic be extended to other F20 quintics?
For the DeMoivre's quintic $x^5 - 5ax^3 + 5a^2x - b = 0$ exist formulas for the relations between the roots. When two roots, say $x_1$ and $x_2$, are known then one can find the other roots with the ...
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Solvability of quintics with complex coefficients?
I was trying to explain Galois theory to a non-specialist and was given a question I couldn't solve to my satisfaction, apologies if I have missed something obvious.
By Galois theory, a polynomial $f(...
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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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Find the minimum value of $\frac {2n^5+n\sqrt [5]{2}+1}{n^2}$ where $n\in\Bbb R^{+}$
Find the minimum value of $$\frac {2n^5+n\sqrt [5]{2}+1}{n^2}$$
where $n\in\Bbb R^{+}$.
I tried to write this expression as $\dfrac {2n^5+n\sqrt [5]{2}+1}{n^2}=a$. From here we have,
$$2n^5-an^2+n\...
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quintic solution to noncommutative polynomial
So, we know there are generally no solutions to an arbitrary quintic polynomial (EDIT from comments: Of course, I mean you can't write the roots using radicals, not that the roots don't exist). I don'...
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Can all the roots of $ax^5+bx^2+c=0$, with real coefficients and $a,c\neq0$, be real numbers?
Let $ax^5+bx^2+c=0$ and $a,c\neq 0$ and $a,b,c$ are real numbers.
Can all the roots of this quintic equation be real numbers?
I divided each side by $a$ and I got
$$x^5+\frac bax^2+\frac ca=0$$
Using ...
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Show that all solutions of $z^5 - z + 16 = 0$ satisfy $1 \lt |z| \lt 2$
I have to show that all solutions of $z^5 - z + 16 = 0$ satisfy $1 \lt |z| \lt 2$.
My attempt: By using Euler's formula I can rewrite the equation into $r^5e^{5i\phi} - re^{i\phi} = -16$ and then ...
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Precise examples for the solvable quintic with the "most complexity" of roots
We know that, the "greater" the degree of a polynomial equation, the greater the "complexity" of the roots in general.
For example, the overall complexity of the roots of a general ...
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How to deduce Abel's theorem from Kronecker's theorem
Kronecker's theorem and Abel's theorem are stated as follow.
Kronecker's theorem: Assume $f(x) \in \mathbb{Q}[x]$ is irreducible over $\mathbb{Q}$ and $\deg{f(x)=p}$ where $p$ is an odd prime number.
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Factoring the quintic $n^5-16n^4+95n^3-260n^2+324n-144$
I was attempting to solve $n^5-16n^4+95n^3-260n^2+324n-144=0$ but then realised I didn't know how to.
How would one go about factoring such a quintic and solve for n? I know that the factored form is ...
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Radical representation of $\cos\frac {2\pi}{11}$
I want to find radical representation of $\cos\dfrac {2\pi}{11}$.
My attempt
Consider the 11th root of unity:
$$
\begin{aligned}
ω&= e^{2πi/11}\\
ω^n&=ω^{n\bmod 11}
\end{aligned}
$$
From Euler'...
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Can a general quintic be solved using Inverse Beta Regularized function?
Tyma Gaidash has recently posted solutions to some quintics in terms of Inverse Beta Regularized function. He also found the closed form for the equation $\cos x=x$ using the same Inverse Beta ...
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Can roots of quintic polynomials be a solution of radicals? [closed]
Assuming integer co-efficients, does there exist a solution to a quintic polynomial that is a solution of radicals? I understand that there is no general formula to solve any quintic, but that doesn't ...
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Finding the number of roots of $f(z) = z^5 + z^3 + 3z + 1$ in the unit disk
Suppose we have $$f(z) = z^5 + z^3 + 3z + 1$$ Find how many roots this function has in the open unit disc $\{z : |z| < 1\}$.
Here's what I think about it:
I tried to split $f$ into two functions $...
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Is there a systematic method to decompose this?
Suppose that
$$x^5+ax^4+bx^3+cx^2+dx+e=0$$
with coef.. all integers, decompose as follows
$$(x^2+Ax+B)(x^3+Cx^2+Dx+E)=0$$
which leads to the following system
$$
\begin{cases}
C+A = a\\
...
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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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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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The most generic radicals-solvable quintic
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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