Linked Questions

56
votes
1answer
6k views

How to solve fifth-degree equations by elliptic functions?

From F. Klein's books, It seems that one can find the roots of a quintic equation $$z^5+az^4+bz^3+cz^2+dz+e=0$$ (where $a,b,c,d,e\in\Bbb C$) by elliptic functions. How to get that?
7
votes
2answers
2k views

Solving 5th degree or higher equations

According to this, there is a way to solve fifth degree equations by elliptic functions. Some related questions that came to mind: Besides use of elliptic functions, what other (known) methods are ...
8
votes
2answers
12k views

How do you solve 5th degree polynomials?

I looked on Wikipedia for a formula for roots of a 5th degree polynomial, but it said that by Abel's theorem it isn't possible. The Abel's theorem states that you can't solve specific polynomials of ...
8
votes
2answers
599 views

Golden ratio, $n$-bonacci numbers, and radicals of the form $\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\cdots}}}$

The following infinite nested radical $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$$ is known to converge to $\phi=\displaystyle\frac{\sqrt{5}+1}{2}$. It is also known that the similar infinite ...
10
votes
0answers
935 views

Deriving the Bring-Jerrard quintic using a $cubic$ Tschirnhausen transformation

One can simultaneously eliminate three terms from the general quintic using a quartic Tschirnhausen transformation. The 4th parameter of the quartic allows it to be done in radicals and details are in ...
2
votes
1answer
706 views

How to transform the general quintic to the Brioschi quintic form?

The quintic can be transformed, in radicals, to the one-parameter Brioschi form, $$w^5-10\alpha w^3+45\alpha^2w-\alpha^2 = 0\tag1$$ Letting $w = 1/(x^2+20)$ and it becomes the rather nice, $$(x^2+...
3
votes
2answers
130 views

How to remove the second two leading terms in the general quintic with just algebra?

Motivated by How to transform a general higher degree five or higher equation to normal form? The goal of the linked question is to transform the general quintic $$x^5+ax^4+bx^3+cx^2+dx+e=0$$ into ...
4
votes
1answer
327 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 ...
1
vote
1answer
235 views

Reduce quintic equation

If we have the general quintic equation $$ax^5+bx^4+cx^3+dx^2+ex+f=0$$ we can vanish the quartic term by doing the substitution $x=y-b/5a$. The question I wanna ask is if there is a possible way to ...
3
votes
1answer
164 views

Is it possible to “depress” any term in a polynomial with a suitable substitution?

If we have a degree $n$ polynomial $$p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x+a_0$$with coefficients in a field, say $\Bbb C$, for concreteness, it is well known that the substitution $y= x + \...
0
votes
1answer
136 views

Substitution to linear + nth power form

Given an arbitrary polynomial: $$a_0 + a_1x + a_2x^2 ... a_nx^n$$ Does there exist a series of substitutions (or single substitution if you choose to combine them) that leaves this function in the ...
1
vote
0answers
119 views

Intuition behind solving the quintic with special functions?

This answer and this wikipedia section describe solving the quintic using The Hermite–Kronecker–Brioschi method. The functions involved seem a bit overpowered, but it is a relatively simple method. I ...
1
vote
2answers
56 views

Is there any general way of calculating polynomial zeroes? [closed]

Is there any way to calculate all zeroes of a $n$-th degree polynomial, like some general formula?
1
vote
0answers
68 views

Reducing the Quintic in the Spirit of Cardano

I have been trying to follow a proof of the Abel Ruffini theorem as described in "Mathematical Omnibus", which starts with a quintic in the Bring Gerrard form. However, the only things that I could ...