Can 5th order equations be solved by means other than radicals?

In summary, the conversation discusses the Abel-Rufini theorem, which states that 5th order equations cannot be solved using radicals. The question is raised about whether there are other analytical methods for solving these equations, aside from numerical solutions. The possibility of using modern or exotic mathematics is mentioned, and a potential solution by Hermite is referenced. This solution can be found in the Bring radical, specifically in the Hermite-Kronecker-Brioschi characterization.
  • #1
meldraft
281
2
As the title suggests, we know from the Abel-Rufini theorem that 5th order equations cannot be solved using radicals. I haven't managed however to find an either positive or negative answer to the following statement:

"Can 5th order equations be analytically solved by means other than radicals?"

This of course rules out numerical solutions. It is pretty contrary to my intuition, however I am entertaining the notion in case it can be solved using modern (or exotic :tongue:) mathematics that are not yet easily found in books.

Any ideas mathematicians? :biggrin:
 
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  • #2
In a history of math book that I read as a teenager, I recall that Hermite produced a non-algebraic solution. So I just now went searching for it on the web, and I think it is this:
http://en.wikipedia.org/wiki/Bring_radical See the section on The Hermite–Kronecker–Brioschi characterization. I haven't studied this material.
 

1. Can 5th order equations be solved using the quadratic formula?

No, the quadratic formula can only be used to solve equations up to 2nd order. For 5th order equations, other methods such as numerical approximation or using specialized functions may be necessary.

2. Are there any alternative methods for solving 5th order equations?

Yes, there are various techniques such as the Newton-Raphson method, the Lagrange resolvent, and the Bring radical that can be used to solve 5th order equations. However, these methods may not always provide exact solutions and may require further simplification.

3. Is it possible to solve 5th order equations without using complex numbers?

Yes, it is possible to solve 5th order equations without using complex numbers. However, the solutions may involve complex numbers depending on the equation and the chosen method of solving.

4. Can computers solve 5th order equations?

Yes, computers can solve 5th order equations using numerical methods. These methods involve approximating the solutions to a desired degree of accuracy using algorithms and computer programming. However, it is still important for scientists to understand the underlying principles and limitations of these methods.

5. Are there any real-life applications of 5th order equations?

Yes, 5th order equations can be used to model various physical phenomena and real-world problems. For example, they can be used to describe the motion of a pendulum, the shape of a vibrating string, and the behavior of certain chemical reactions. They are also commonly used in fields such as engineering, physics, and economics.

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