5th degree polynomial equations

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5th degree polynomial equations

Postby Kurushimi » Thu Jun 10, 2010 1:23 am UTC

I was just thinking. It was proven by Galois that the solutions to equations of 5th degree polynomials and above could not, in general, be written in terms of the square root/ cube root / nth root functions. So, I was wondering. Does there exist a function such that the roots of the polynomial equations of a particular degree can be written in terms of it? Can a "class" of functions (like the class of nth root functions) be used in general for any polynomial of any degree?

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Re: 5th degree polynomial equations

Postby skeptical scientist » Thu Jun 10, 2010 2:33 am UTC

Quintics can be solved with radicals and Bring radicals. I'm not sure about generalizations to higher degrees.
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Re: 5th degree polynomial equations

Postby imatrendytotebag » Thu Jun 10, 2010 6:28 am UTC

The proof that a 5th degree polynomial cannot be solved by radicals relies on the fact that the permutation group S5 is not solvable, meaning we cannot have a chain of subgroups {e} = G0 < G1 < ... < Gn = S5 such that each group is normal in the next and the quotient groups are all cyclic. This is because (and I am somewhat oversimplifying here) that the operation of taking an nth root is effectively adding a cyclic extension to a field.

To be slightly more precise, if we have a field F, some element a in F, and want to add to F the nth root of a, we will get some field K containing F and the nth root of a, and the group of automorphisms of K that fix F is cyclic. (This is not exactly true, we need F to have all the nth roots of unity also).

So if, in the language of Galois theory, adding an nth root <=> a cyclic extension, perhaps we need some class of polynomials such that adding a root of a polynomial in that class <=> some other kind of group type extension. Bring radicals talk about polynomials in the class x^5 + x + a, but maybe a more general class of polynomials would be more suitable...

Anyway. Let me know if this makes any sense.
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