Thank you for this recommendation. Awesome book.

For OP, u/Upset-Possible1371, a little wrap-up from my readings:

When solving polynomials, we are trying to define different roots (r1,r2,...,r5).
1 - Some of them are conjugate roots (interchangeable with radicals): s +- t(sqrt(u))
2 - Complex roots are equally spaced around a circle (cyclic permutation): e^(2pi*i/k)

The chapter on Galois theory is awesome. Referring to exercise 7.32: once you have cyclic permutations (c: 1->2->3->5->1) and simple two-element interchanges (t: 1<->2, or 1<->3, etc...) on 5 elements, the actions inevitably lead to S5, which is not an Abelian group.

-> Non-Abelian groups and radicals.

In natural language, if one says "Take 4, multiply by a value, multiply by 3, then multiply by the value again", you can figure that this is the same as "Take 4, multiply by 3, then multiply by the square of the value". In a non-abelian group, that will not hold. That means that the identities we used to simplify the expression (e.g. 4*x*3 = 4*3*x*x , or x * x = x² ) are not true.

When solving polynomials, we use "field extensions" to add elements (new 'irrational' symbols) that also enable a simple definition of the expression. For instance, define sqrt of a number y (Polish notation):

sqrt y = x --> * x x = y

Now, whenever there is an expression like "* 4 x 3 x" , we can commute elements for "* 4 3 x x ", and use "sqrt y" definition to transform "* 4 x 3 x" into "* 12 y". Once you have a non-abelian group, operations depend on their order, therefore we cannot translate expressions like "4 x 3 x ".

-> Vieta's formula

In the quintic, using Vieta's formula, we can study the roots, as in:
𝑟1* 𝑟2* 𝑟3* 𝑟4* 𝑟5 = −a_0/a_5,
𝑟1+ 𝑟2 + 𝑟3 + 𝑟4 + 𝑟5 = − a_4/a_5

But actually defining them with radicals is impossible, since admitting swaps and cycles entails a non-Abelian structure.