I have read the paper A view of mathematics by Alain Connes. If you read page 28 of this paper, you will read the following paragraph:
Abel and Galois analyzed the symmetries of functions of roots ofpolynomial equations and Galois found that a function of the roots isa "rational" expression iff it is invariant under a specific group $G$of permutations naturally associated to the equation and to the notionof what is considered as being "rational". Such a notion defines afield $K$ containing the field $\Bbb Q$ of rational numbers alwayspresent in characteristic zero. He first exhibits a rational function$$V(a,b,\cdots,z)$$ of the $n$ distinct roots $(a,b,\cdots,z)$ of agiven equation of degree $n$, which affects $n!$ different valuesunder all permutations of the roots, i.e. which "maximally" breaksthe symmetry. He then shows that there are $n$"rational" functions$\alpha(V),\beta(V),\cdots$ of $V$ which give back the roots$(a,b,\cdots,z)$. His group $G$ is obtained by decomposing inirreducible factors over the field $K$ the polynomial (over $K$) ofdegree $n!$ of which $V$ is a root. Using the above rational functions$$\alpha(V_j), \beta(V_j), \cdots$$ applied to the other roots $V_j$of the irreducible factor which admits $V$ as a root, yeilds thedesired group of permutations of the $n$ roots $(a,b,\cdots,z)$.
I have known that Galois theory is rather a theory that describes symmetry, but this paper explains that Galois theory is a theory that came about because it shattered symmetry which is contradictory, so I was confused.
So, I would appreciate it if you could explain in detail and in simple terms what is meant by "maximally" breaks the symmetry in this paper.
