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Can someone knowledgeable about this chime in: > The invention of group theory. In proving that there are no general algebraic solutions for the roots of quint

Can someone knowledgeable about this chime in: > The invention of group theory. In proving that there are no general algebraic solutions for the roots of quintic equations, Abel invented (independently of Galois) what later became known as group theory. In addition to Galois, the topic was also studied in the same period by Joseph-Louis Lagrange (1736–1813). How are quintic equations related to group theory?

cloudvertigo 2y ago
You may want to look up Galois Theory. The core idea is to study the equations' roots via their permutation group. If and only if the permutation group of the roots (Galois group) is a Solvable group, the equation has algebraic solutions.
eigenket 2y ago
The connection is a field of study today called Galois theory, and especially the "fundamental theorem of Galois theory" https://en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_theory https://en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_... Roughly speaking what you do is you start with a polynomial over some field, for example over the rational numbers, then you see what you need to add to the rational numbers to get to a field in which you can fully factor that polynomial into linear factors. For example say we have the polynomial x^2 - 2, we know there isn't any solution to this in the rationals, so we can't factor the polynomial. We then consider the expanded field you get when you add the square root of 2 to the rationals. This expanded field includes root 2, and all products and sums of root 2 with rational numbers. You can check that the elements of this new field look like a + b sqrt(2) where a and b are rational. In this new field you can factor the above polynomial as (x + sqrt(2))(x - sqrt(2)). The connection with group theory comes when you realise that the central object of your study is the bijective (invertable) functions which map this new extended field to itself, while mapping the rationals to themselves. For example for the field formed from the rationals by adding root 2 there are two such bijective functions: the identity function which maps everything to itself, and a second one which swaps root 2 with minus root 2, and leaves everything else the same. This jump to having to think about these groups of functions (automorphism groups) is a big imaginative leap, but let's you turn hard problems about polynomials into easier problems about groups
deleted 2y ago
[deleted]
vismit2000 2y ago
https://news.ycombinator.com/item?id=41255456 https://news.ycombinator.com/item?id=41255456
rramadass 2y ago
https://math.stackexchange.com/questions/2067201/need-help-understanding-the-relation-between-galois-theory-and-a-general-quintic https://math.stackexchange.com/questions/2067201/need-help-u...

cloudvertigo 2y ago

You may want to look up Galois Theory. The core idea is to study the equations' roots via their permutation group. If and only if the permutation group of the roots (Galois group) is a Solvable group, the equation has algebraic solutions.

eigenket 2y ago

The connection is a field of study today called Galois theory, and especially the "fundamental theorem of Galois theory" https://en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_theory https://en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_... Roughly speaking what you do is you start with a polynomial over some field, for example over the rational numbers, then you see what you need to add to the rational numbers to get to a field in which you can fully factor that polynomial into linear factors. For example say we have the polynomial x^2 - 2, we know there isn't any solution to this in the rationals, so we can't factor the polynomial. We then consider the expanded field you get when you add the square root of 2 to the rationals. This expanded field includes root 2, and all products and sums of root 2 with rational numbers. You can check that the elements of this new field look like a + b sqrt(2) where a and b are rational. In this new field you can factor the above polynomial as (x + sqrt(2))(x - sqrt(2)). The connection with group theory comes when you realise that the central object of your study is the bijective (invertable) functions which map this new extended field to itself, while mapping the rationals to themselves. For example for the field formed from the rationals by adding root 2 there are two such bijective functions: the identity function which maps everything to itself, and a second one which swaps root 2 with minus root 2, and leaves everything else the same. This jump to having to think about these groups of functions (automorphism groups) is a big imaginative leap, but let's you turn hard problems about polynomials into easier problems about groups

deleted 2y ago

[deleted]

vismit2000 2y ago

https://news.ycombinator.com/item?id=41255456 https://news.ycombinator.com/item?id=41255456

rramadass 2y ago

https://math.stackexchange.com/questions/2067201/need-help-understanding-the-relation-between-galois-theory-and-a-general-quintic https://math.stackexchange.com/questions/2067201/need-help-u...