The connection is a field of study today called Galois theory, and especially the "fundamental theorem of Galois theory" https://en.wikipedia.org/wiki/Fundamen

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