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Buzz Bloom

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I became curious about the following problem from a discussion in another thread:

After a bit of study I concluded that the meaning of the assertion below regarding some specific real number rl P has the meaning which follows it.

The article

contains the following text:

My QUESTION IS:

Any help would be much appreciated.

After a bit of study I concluded that the meaning of the assertion below regarding some specific real number rl P has the meaning which follows it.

Assertion: "r is not expressible in terms of radicals."

Meaning: r is not expressible in terms of a finite application of a collection of operators (+, -, ×, and /, together with any of the n-th roots (where n a positive integer) ), where these operators are applied to integer operands, or to expressions of the same kind.

If this meaning is incorrect, I hope someone will correct it.Meaning: r is not expressible in terms of a finite application of a collection of operators (+, -, ×, and /, together with any of the n-th roots (where n a positive integer) ), where these operators are applied to integer operands, or to expressions of the same kind.

The article

contains the following text:

An example of a quintic whose roots cannot be expressed in terms of radicals is

This simple example of a Bring-Jerrard quintic equation has one real root, with an approximate value of*x*^{5}−*x*+ 1 = 0.r = -1.16730397783.

This can be seen in the attached PNG file.My QUESTION IS:

What would a proof that "r is not expressible in terms of radicals" look like?

I have no idea whatever how one would go about proving the non-radical nature of just this one example.Any help would be much appreciated.