Intuition behind looking at permutations of the roots in Galois theory

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:

1. 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?

2. 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,

First, remember that "solvability of a polynomial" had a very specific meaning: it meant finding the roots as expressions of the coefficients, and specifically expressions involving only the basic operations of additions (including subtraction), multiplication (including division) and extraction of radicals. For example, the solutions to the linear and quadratic equation are of this sort, since the roots of $p(x)=ax+b$ can be expressed as $-b/a$; the roots of $p(x)=ax^2+bx+c$ can be expressed as $\frac{-b+\sqrt{b^2-4ac}}{2a}$ and $\frac{-b-\sqrt{b^2-4ac}}{2a}$, etc. Similar with cubics and biquadratics.

Why are permutations of the roots interesting? You want to remember that if you have a monic polynomial, then the coefficients are symmetric functions of the roots: $$(x-r_1)(x-r_2)\cdots(x-r_n) = x^n -(r_1+\cdots+r_n)x^{n-1} + \cdots + (-1)^n(r_1\cdots r_n).$$ These coefficients are such that if you permute the roots, the coefficients don't change.

If we define the elementary symmetric functions on $r_1,\ldots,r_n$ as follows: $$\begin{align*} s_0(r_1,\ldots,r_n) &= 1\\ s_1(r_1,\ldots,r_n) &= r_1+\cdots + r_n\\ s_2(r_1,\ldots,r_n) &= r_1r_2 + r_1r_3 + \cdots + r_1r_n + r_2r_3 + \cdots + r_{n-1}r_n\\ &\vdots\\ s_n(r_1,\ldots,r_n) &= r_1\cdots r_n; \end{align*}$$ that is, $s_i(r_1,\ldots,r_n)$ is the sum of all possible products of $i$ distinct roots; then we have $$(x-r_1)\cdots(x-r_n) = x^n + (-1)s_1(r_1,\ldots,r_n)x^{n-1} + \cdots+ (-1)^n s_n(r_1,\ldots,r_n).$$

Suppose that $\sigma$ is a permutation of $\{1,\ldots,n\}$. If $\mathbb{Q}[x_1,\ldots,x_n]$ is the set of all rational polynomials in $n$ variables, then $\sigma$ acts on $\mathbb{Q}[x_1,\ldots,x_n]$ by mapping $p(x_1,\ldots,x_n)$ to $p(x_{\sigma(1)},\ldots,x_{\sigma(n)})$. We can then ask: what is the subset of $\mathbb{Q}[x_1,\ldots,x_n]$ that is fixed pointwise by the action of $S_n$? Clearly, the elementary symmetric functions are fixed pointwise; so are others. For instance, $x_1^2+\cdots + x_n^2$ is fixed pointwise.

The polynomials that are invariant under the action of $S_n$ are the "symmetric functions" on $x_1,\ldots,x_n$. Newton proved that the elementary symmetric functions generate the symmetric functions: every symmetric function can be expressed as combination of the elementary symmetric functions.

So the coefficients of a polynomial are intimately related to the symmetric functions of the roots, which are in turn intimately connected with the action of $S_n$ on the roots.

For example, let's consider the quadratic equation in this light. We have $$x^2 +bx + c = (x-r_1)(x-r_2),$$ so $b=-(r_1+r_2)$, $c=r_1r_2$. To express $r_1$ and $r_2$, separately, using $b$ and $c$, consider the symmetric polynomials $(r_1+r_2)^2$ and $(r_1-r_2)^2$ on the roots. Since these are symmetric, they can be expressed in terms of $b$ and $c$ (which generate the symmetric polynomials). Indeed, $$\begin{align*} (r_1+r_2)^2 &= (-(r_1+r_2))^2 = b^2;\\ (r_1-r_2)^2 &= (r_1+r_2)^2 - 4r_1r_2 = b^2 - 4c. \end{align*}$$ Thus, $|r_1-r_2| = \sqrt{b^2 - 4c} $ , so $r_1-r_2 = \sqrt{b^2-4c}$ or $r_1-r_2=-\sqrt{b^2-4c}$. Since $r_1+r_2 = -b$, we have $$\begin{align*} r_1 &= \frac{1}{2}\Bigl( (r_1+r_2) + (r_1-r_2)\Bigr) =\left\{\begin{array}{l} \frac{1}{2}\Bigl( -b +\sqrt{b^2-4c}\Bigr)\\ \text{or}\\ \frac{1}{2}\Bigl( -b -\sqrt{b^2-4c}\Bigr) \end{array}\right.\\ r_2 &=\frac{1}{2}\Bigl( (r_1+r_2) - (r_1-r_2)\Bigr) = \left\{\begin{array}{l} \frac{1}{2}\Bigl( -b - \sqrt{b^2-4c}\Bigr)\\ \text{or}\\ \frac{1}{2}\Bigl( -b+\sqrt{b^2-4c}\Bigr) \end{array}\right. \end{align*}$$ which gives the usual quadratic formula: one root of $x^2+bx+c$ equals $\frac{-b+\sqrt{b^2-4c}}{2}$, the other equals $\frac{-b-\sqrt{b^2-4c}}{2}$.

A similar approach can be used for the cubic and the biquadratic. The question is whether something similar can be done with the quintic and higher. This particular straightforward approach (originally due to Lagrange) runs into a problem: to solve a cubic, you end up having to solve a quadratic equation on the elementary symmetric polynomials. To solve a biquadratic, you end up having to solve a cubic. But to solve a general quintic, you end up having to solve a polynomial of degree six! So you run into a roadblock.

Galois Theory studies the roots by studying the "symmetries" among the roots, by considering their permutations (which necessarily leave the coefficients fixed), and considering how certain subgroups of $S_n$ leave (or not) the coefficients or other functions of the roots fixed.

As to your second question: the group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ has, as a quotient, every single Galois group over $\mathbb{Q}$. This is known to include at least all the solvable groups (a theorem of Shafarevich), as well as many of the nonabelian simple groups. It is still an open question whether every group is in the Galois group of a polynomial over $\mathbb{Q}$. The group itself can be described abstractly, but we still don't have a very good "feel" for it. In fact, a lot of the work on Galois representations (which was key to the proof of the Taniyama-Shimura Conjecture) has to do with understanding "just" the image of this group in suitable matrix groups (i.e., trying to understand the representation theory for this group, in order to gain some insight into the group itself).

As for "intuition": any infinite Galois extension is completely determined by its finite Galois sub-extensions; this is why $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ is a profinite group: it is determined by the Galois groups of the finite sub-extensions it has as quotients. The possible images of an element $a\in\overline{\mathbb{Q}}$ under any homomorphism $\overline{\mathbb{Q}}\to\overline{\mathbb{Q}}$ that fixes $\mathbb{Q}$ must be another root of the minimal polynomial of $\alpha$, and so the homomorphism will restrict to an automorphism of the Galois closure of $\mathbb{Q}(\alpha)$. Any element of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ is thus determined by its action on these finite Galois extensions, and the inverse limit is a way of "gluing" all this information together in a coherent way. But the group is far from understood.

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)

1. LA PENSEE D'EVARISTE GALOIS ET LE FORMALISME MODERNE
2. The Ideas of Evariste Galois:Recovering Motivation in Abstract Algebra Through the Exploration of Original Sources
3. Galois and his groups (and references therein)
4. Original Works of Evariste Galois, on math.SE (and references)
5. The (not so) simple idea of Galois Theory, on scribd
6. GALOIS THEORY AFTER GALOIS
7. 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:

1. Fields containing $n$ indeterminates can be "symmetrized".
2. The Galois group of a radical extension is solvable.
3. 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.