TITLE: Intuition behind looking at permutations of the roots in Galois theory
QUESTION [44 upvotes]: What I find after reading books is that they explain only the conceptual definition and no one mentions the explanation behind it;
I have been reading the Galois theory as many people told me to read,
so I am wondering:

Why are permutation of roots important, and how do they influence the solvability of a polynomial?
Can anyone briefly explain the direct intuition behind why did Galois consider the permuations?

Please tell me: what does the profinite group $\mathrm{Gal}(\mathbb{\bar{Q}}/\mathbb{Q})$ contain?
Is there any logical intuition behind that, like I want the answer in the form, that if someone asks what do homology sequence account for, its intuition can be given as "it counts the holes that map to another surface".


After reading about Galois theory, I find that I understood everything, but there are large blocks that stumble me.
Please help me in understanding intuition behind the theory.
Thanks a lot every one,

REPLY [2 votes]: To give another analysis (on the already fine answer):
The intuition behind Galois Theory, is that in order to find solutions based on radicals (polynomial functions of the coefficients), one has to do it in a consistent step-by-step manner.
In Galois theory one progresively constructs group extensions of a polynomial equation in a step-by-step (or root-by-root) manner. But since the roots are unknown beforehand how do we start and how do we continue?
The answer is that the ordering of the roots (or group extensions associated with each root) should play no part in the process, thus a unique un-ambiguous result can be obtained (consistent method).
As mentioned this consistency has to do with permutations, meaning if another ordering of the steps is followed, the result can be unique and un-ambiguous (the ordering of the steps).
This means that studying the permutations of the roots which leave the (field of) coefficients of the polynomal equation fixed (Galois group extensions) is important in this study.
This is translated in modern Galois Theory as the sentence: "The Galois group extensions are solvable", thus the polynomial equation is solvable by radicals. But Galois (actually Artin formulated the modern version of the theorem) found that the Galois group of an arbitrary quintic does not have this property (in the sense of the present analysis this means that the structure of general quintics does have ambiguity in the process of costructing galois extensions in a step-by-step or root-by-root manner)
References (mostly related to the original Galois memoir and historical evolution)

LA PENSEE D'EVARISTE GALOIS ET LE FORMALISME MODERNE
The Ideas of Evariste Galois:Recovering Motivation in Abstract Algebra Through the Exploration of Original Sources
Galois  and  his  groups (and references therein)
Original Works of Evariste Galois, on math.SE (and references)
The (not so) simple idea of Galois Theory, on scribd
GALOIS THEORY AFTER GALOIS
Galois Theory for Beginners 

Specificaly the last reference (7) makes explicit the previous analysis and derives the central result of Galois using basic abstract algebra:

The aim of this paper is to prove the unsolvability by radicals of the
quintic (in fact of the general $n$th degree equation for $n \ge 5$)
using  just the fundamentals of groups, rings and fields from a
standard first course in algebra. The main fact it will be necessary
to know is that if $\phi$ is a homomorphism of group $G$ onto group
$G'$ then $G' \sim G / \ker \phi$, and conversely, if $G/H \sim G'$
then $H$ is the kernel of a homomorphism of $G$ onto $G'$. The concept
of Galois group, which guides  the whole proof, will be defined when
it comes up. With this background, a proof of unsolvability by
radicals can be constructed from just three basic ideas, which will be
explained more fully below:

Fields containing $n$ indeterminates can be "symmetrized".
The Galois group of a radical extension is solvable.
The symmetric group $S_n$ is not solvable.


Important side-Note
In order to acquire intuition of why certain results or approaches or formulations were used (or are taught today), one has to study the historical evolution of a certain problem, how it started, what triggered it, and the various approaches followed in the course of time.
Usually, this aspect is rarely (if ever) mentioned in textbook courses or lectures and most of the time a ready-made result is presented leaving the student in a sense of magical-out-of-the-hat result which probably requires magic powers or an unexplained genius to comprehend it.
But following the historical evolution the student comes in contact with the first occurance of a problem and what triggered it. This in itself provides enough intuition behind a mere formalistic formulation. Furthermore, one sees recurring paterns in approaches or variations of the same approach applied (this provides a unified view of results and methods that would seem totaly un-related otherwise).
Finaly one should bare in mind that all this historical knowledge (and scientific publications and approaches) are available in prestige university libraries and institutions (and their residents and employees) while usualy not available in a student on his own study. This also explains why certain modern breakthroughs in science were later(?) found to correlate to past or ancient papers or scientific approaches which are only available to those having access to institutions like British Library (or similar). Not to mention independent discovery/invention of solution to a problem by several different people on different times and places, while only one of them will be made known to the general audience.