I am seeking to learn about the motivation in the development of group theory. It has been a few years since algebra, and we got as far as rings and fields. I am aware that there were several motivations for the development of the theory of groups; the studies of different geometries, the groups that arose out of the study of modular arithmetic, but i am particularly interested in the motivation from the quest to disprove the solvability of the quintic equation, generally, by radicals.
I understand after the solutions for the lower order polynomial functions were discovered it was believed that there would be no like solution for that of quintic equations. I have read of the work Lagrange did that led Abel to develop his proof. Particularly the link on the history of the development of group theory provides some of the history.
My specific question relates to the discussion of the role the permutations of the roots the 5th order polynomials played. as follows from the above link
Lagrange assumes the roots of a given cubic equation are x', x'' and x'''. Then, taking 1, w, w2 as the cube roots of unity, he examines the expression
$$R = x' + wx'' + w2x'''$$
and notes that it takes just two different values under the six permutations of the roots x', x'', x'''
If R is a polynomial, and the roots are as denoted above, what are these cube roots of unity? What is the interpretation of the permutation of the roots, and their relevance to a solution by radicals?
I have read the article on the Abel-ruffini theorem, which provides a proof which is based on Galois theory. What i would like is more of an explanation in terms of the mathematics the problem was understood in during the time it was posed. i.e. Not the modern algebraic formulation it is introduced in today. I have not studied Galois theory, and considering my own limitations, perhaps the context it was understood in at the time would be more accessible to my understanding.
Also, i did try to reason through Intuition behind looking at permutations of the roots in Galois theory, and the part that was difficult was the discussion of elementary symmetric functions.
UPDATE
A paper entitled Galois theory for beginers was submitted in an answer below. I would like to include an excerpt from that paper that is more direct at part of my difficulty.
the set of elements obtainable from $a_0,\ldots,a_{n-1}$ by +, -, $\times$, $\div$ is the field $\mathbb{Q}(a_0,\ldots,a_{n-1})$. If we denote the roots of $(\ast)$ by $x_1,\ldots,x_n$, so that $$(x-x_1)\cdots(x-x_n)=x^n=a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$ then $a_0,\ldots,a_{n-1}$ are polynomial functions of $x_1,\ldots,x_n$ called the elementary symmetric functions: $$a_0=(-1)^nx_1x_2\ldots x_n,\ldots,a_{n-1}=-(x_1+x_2+\cdots+x_n).$$
the link to the paper is provided in a comment below the answer. what i understand that the field $\mathbb{Q}$ is formed from the coefficients of the nth order monic polynomial, $a_0,\ldots,a_{n-1}$, closed under the operations of addition and multiplication.
But in what sense are these coefficients a polynomial of the roots of the polynomial? Also, what is the meaning of the equation $a_0=(-1)^nx_1x_2\ldots x_n,\ldots,a_{n-1}=-(x_1+x_2+\cdots+x_n)$?
Is this equation some how related to the identities relating the sum and multiplies of roots of an nth order polynomial to the quotients of the coefficients, $\alpha+\beta=\frac{-b}{a}$ and $\alpha \beta= \frac{c}{a}$ for the second order polynomial with roots $\alpha,\beta$ and coefficients a,b,c?
I think there is a very short but expository article on the topic you are considering. The proof, that a general quintic can not be solved by radicals, do not requires too much theory of field extensions, and other things, as usually seen in books. This is in fact stated in the article below (I hope you may have access to it).