Has it been proven that quintic equations cannot be solved by *any* formula?

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In summary, the Bring radical may be able to solve the quintic equation, but it's an infinite series.
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It seems that Abel's theorem says that the quintic cannot be solved by arithmetic & root operations, but couldn't there be the situation where another function is used in concert with these operations?
 
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swampwiz said:
It seems that Abel's theorem says that the quintic cannot be solved by arithmetic & root operations, but couldn't there be the situation where another function is used in concert with these operations?

If you restrict yourself to finite expressions, yes, that is true. However, if you allow such things as infinite series and the like, you can solve quintic equations---in terms of hypergeometric functions. See, eg.,
http://mathworld.wolfram.com/QuinticEquation.html
 
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OK, so it looks like the Bring radical could work, but it's an infinite series.
 
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swampwiz said:
It seems that Abel's theorem says that the quintic cannot be solved by arithmetic & root operations, but couldn't there be the situation where another function is used in concert with these operations?

Through Galois theory, it is proven that the general solution of a polynomial of degree at least 5 is not expressible in terms of the operations addition, multiplication (and their inverses) and taking roots.

Other ways are still possible.
 
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Historically, mathematicians would compete for solving various types of polynomials. Tartaglia, an amateur Italian mathematician came up with the general formula for roots of a cubic and created a poem encoding the formula to prevent others from claiming they found it first.

https://www.storyofmathematics.com/16th_tartaglia.html

From there Ferrari, another younger mathematician conquered the quartics and the quintics remained unsolvable until Abel definitively proved they were by Galois theory.
 
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It depends on whether the associated Galois group of the roots is a solvable group.
 
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  • #10
And we know both that, of course there are nonsolvable groups and that these are the Galois groups of some polynomials.
 
  • #11
of course "roots" are also given by infinite series.
 

1. What is a quintic equation?

A quintic equation is a polynomial equation of the fifth degree, meaning it has a highest degree of 5. It can be written in the form ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0, where a, b, c, d, e, and f are constants and x is the variable.

2. Can all quintic equations be solved?

No, not all quintic equations can be solved. Some quintic equations have solutions that can be expressed using radicals, while others cannot be solved by any formula. This is known as the Abel-Ruffini theorem.

3. What is a formula for solving quintic equations?

There is no general formula for solving all quintic equations. However, there are specific formulas for solving certain types of quintic equations, such as depressed quintic equations or symmetric quintic equations.

4. How do we know that quintic equations cannot be solved by any formula?

This was proven by mathematicians Paolo Ruffini and Niels Henrik Abel in the early 19th century. They showed that there is no general formula for solving quintic equations using only the four basic arithmetic operations (addition, subtraction, multiplication, and division) and taking roots.

5. Are there any other methods for solving quintic equations?

Yes, there are other methods for solving quintic equations, such as numerical methods and approximations. These methods involve using computers or calculators to find approximate solutions to the equation. However, there is no exact formula for solving all quintic equations.

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