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Comment by gizmo686 | Hacker News Reader

> Essentially, it means the number cannot be constructed using a finite combination of integers and standard algebraic operations (addition, subtraction, multiplication, division, and integer roots)

Slight clarification, but standard operations are not sufficient to construct all algebraic numbers. Once you get to 5th degree polynomials, there is no guarantee that their roots can be found through standard operations.

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I am no mathematician, but i think you may be overstating Galois result. it says that you cant write a single closed form expression for the roots of any quintic using only (+,-,*,/,nth roots). This does not necessarily stop you from expressing each root individually with the standard algebraic operations.

I think you are thinking of the Abel–Ruffini impossibility theorum, which states that there is no general solution to polynomials of degree 5 or greater using only standard operations and radicals.

Galois went a step further and proved that there existed polynomials whose specific roots could not be so expressed. His proof also provided a relatively straightforward way to determine if a given polynomial qualified.

Thanks for the correction. It seems that all the layman’s explanations on Galois theory i have seen have been simplified to the point of being technically wrong, as well as underselling it.

Technically, the actual statement in Galois theory is even more general. Roughly, it says that, for a given polynomial over a field, if there exists an algorithm that computes the roots of this polynomial, using only addition, subtraction, multiplication, division and radicals, then a particular algebraic structure associated with this polynomial, called its Galois group, has to have a very regular structure.

So it's a bit stronger than the term "closed formula" implies. You can then show explicit examples of degree 5 polynomials which don't fulfill this condition, prove a quantitative statement that "almost all" degree 5 polynomials are like this, explain the difference between degree 4 and 5 in terms of group theory, etc.

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