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(I hope that this question is acceptable and within the rules of math.stackexchange. If not, mods should edit at will and let me know if this question must be broken into several different questions. I ask these all together at once because they seem crucially tied together insofar as answers would correct my misunderstandings.)

I am currently studying Galois Theory, but I am unable to get a handle on the subject. My intuition leads to me conclusions which are obviously incorrect, so I will ask a brief series of questions which I think will help me correct my course.

Let $Q$ be the rationals. Let $f$ be an irreducible polynomial (hence separable) over $Q$ with degree $n$. Let $F$ be the splitting field of $f$. So $F/Q$ is Galois. Let $G$ be the Galois group of $F/Q$. Let $a_1,\dots ,a_n$ be the distinct roots of $f$. My understanding is that $F=Q(a_1,\dots ,a_n)$.

Question 1: Since every automorphism of $F/Q$ permutes $a_1,\dots ,a_n$, it is clear that $G$ can be embedded into $S_n$ as a subgroup. Why is it not the case that $G$ is automatically all of $S_n$? Surely any permutation of the roots gives an automorphism of $F$ preserving $Q$? If not, what might be an instructive minimal example?

Question 2: Conceptually speaking, what exactly prevents certain permutations from being acceptable automorphisms of $F/Q$?

Question 3: Alternatively, if $f$ is not irreducible, then $F$ is the splitting field of some polynomial which is not irreducible. In this case, I believe that roots from different irreducible components cannot jump to roots of other irreducible components. Why is this the case?

Question 4: Again we assume that $f$ is irreducible. What must be unique about the situation in order for $G$ to really be all of $S_n$?

Question 5: Now set $n=4$. I know that $A_{4}$ is the only subgroup of $S_4$ with order $12$. Suppose that $G=S_4$. Suppose $K=Q(a_1)$. Why is it not the case that $F/K$ has order $12$ and hence has Galois group $A_4$? It seems that the Galois group of $F/K$ could include all permutations of $a_1,\dots ,a_4$ that map $a_1$ to itself.

Question 6: Suppose we are in the case of Question 5. Why is it the case that the Galois group of $F/K$ has a transposition?

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  • $\begingroup$ Just note that $|G|\le n$, for an arbitrary field extension $F/Q$ (not necessarily Galois), and that $|S_n|=n!$. For Question 3, if $\sigma$ is an automorphism fixing $Q$, and $f$ is an irreducible polynomial over $Q$ with a root $\alpha$, then $$f(\sigma(\alpha))=\sigma(f(\alpha))=0.$$ $\endgroup$
    – user735332
    Aug 5, 2020 at 3:46
  • $\begingroup$ @chenrk I'm not sure if this is true. I don't think $G$ necessarily needs to have order bounded by $n$. For example, $G$ might be $S_n$. $\endgroup$
    – LAGC
    Aug 5, 2020 at 4:41
  • $\begingroup$ I misunderstood something. By the above $n$ I mean the degree of field extension. In your context, it is indeed possible. $\endgroup$
    – user735332
    Aug 5, 2020 at 5:19
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    $\begingroup$ For Question 1, it may help to think about the fact that your argument doesn't use irreducibility anywhere, so it would apply to something like $f(x) = (x-3)(x-4)$ but clearly there is no automorphism taking $3$ to $4$. After all, just because you have an embedding into $S_n$ is not really evidence that every permutation can be realized as a field automorphism. $\endgroup$
    – user208649
    Aug 5, 2020 at 5:25

4 Answers 4

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Question 1: Consider $f(x) = x^4 + 1$, which is irreducible over $\mathbb{Q}$. Then we may write all the roots of $f(x)$ as $\zeta_8^i$ where $i = 1, 3, 5, 7$. In particular $[F : \mathbb{Q}] = [\mathbb{Q}(\zeta_8) : \mathbb{Q}] = 4$. Thus the galois group cannot be all of $S_4$ - if you write out the automorphisms you will see that saying where $\zeta_8$ goes is the same as saying where all the $\zeta_8^i$ go.

Question 2: This was answered above, sometimes there is dependencies between the roots, so because each element of the galois group is a field automorphism, some will be disallowed.

Question 3: If $F$ is the splitting field of $f(x)$ and $f(x) = g(x)h(x)$ then if $\sigma \in Gal(F / \mathbb{Q})$ we have $\sigma(g(x)) = g(\sigma(x))$. In particular if $\alpha$ is a root of $g(x)$ then so is $\sigma(\alpha)$.

Question 4: This is a somewhat difficult question to answer. There are many criteria for when a subgroup of $S_n$ might be the symmetric group (e.g., if it is a transitive subgroup containing an $(n-1)$-cycle and a transposition). In general if I were trying to answer this in some situation I would do some algebraic number theory and look mod $p$.

Question 5: The degree $[\mathbb{Q}(a_i): \mathbb{Q}] = 4$ ($a_i$ a root of an irreducible polynomial of degree $4$, in particular $[F : \mathbb{Q}(a_i)] = 24/4 = 6$ by the tower law. You are correct, the elements are the permutations that fix $a_i$, but that is just $S_3$!.

Question 6: The element fixing $a_1$ and $a_2$ but permuting $a_3$ and $a_4$ is in $Gal( F / \mathbb{Q}(a_1))$ and is clearly a transposition.

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It's important that the automorphism do more than merely move the roots around, they must be algebraically indistinguishable. Say I start with the base field of the rationals and I extend to include the solutions to $x^2+1$ which has the complex roots $\pm i$ and we can now write all numbers in this field in the form $a + bi$ in $\mathbb{Q}[i]$. I could however set $j=-i$ and then written all the numbers in the form $a + bj$ without changing anything. Complex multiplication works exactly the same way for $j$ because $j^2=-1$ just like it does for $i$.

Now compare this to field $\mathbb{Q}[\sqrt{2},\sqrt{3}]$. Lets use $j= \sqrt{2}$ and $k=\sqrt{3}$ for this example so that $j^2=2$ and $k^2=3$. If I take numbers of the form $a + bj$ and $a + bk$ then the multiplcation doesn't work the same. To see this compare $$(a+bj)(c+dj)=ac+2bd +(bc+ad)j \\ (a+bk)(c+dk)=ac+3bd +(bc+ad)k$$ and so we can see the that if I replace $j$ with $k$ the multiplication behaves differently and so this permutation is not an automorhphism. There are two automorhphisms related to sign switching as in the previous example. I can't tell $\sqrt{2}$ and $-\sqrt{2}$ apart in the above equations in the same way as in the first example and similarly for $\sqrt{3}$ and $-\sqrt{3}$. This means we have identified two non-trivial elements and they have order two. The group must have at least one other element as $2$ does not divide $3$ and these will be related to $jk = \pm \sqrt{6}$ being indistinguishable. Number in this field are all of the form $a + bj + ck + d(jk)$ and any other permutation will change the multiplication so the Galois group is the Klein four group. Permuting some of the roots gave us automorphisms but others changed the multiplication so they weren't automorphisms.

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Question 1 and 2:

A $\mathbb Q$-homomorphism does not only permute the roots of some polynomial. It's also a field automorphism, and it has to obey the axioms of a field automorphism. This may restrict the choices of permutations if the roots can be written as polynomials of each other, since homomorphisms have very strong restrictions when applied to polynomial expressions. For instance, if $\alpha$ is one root of $f$, and the others are $\alpha^2,\dots,\alpha^n$, then the image of that one single root $\alpha$ under a $\mathbb Q$-homomorphism $\sigma$ already determines the images of all the other roots as well, since $\sigma(\alpha^k)=\sigma(\alpha)^k$, removing a lot of choice. An example of such a polynomial is $X^4+X^3+X^2+X+1$, whose roots are $\zeta:=e^{2\pi i/5}$ and $\zeta^2,\dots,\zeta^4$, the primitive 5th roots of unity.

Question 3:

A $\mathbb Q$-automorphism maps a root of any polynomial to another root of the same polynomial, you already got that. If a polynomial can be factorized, let's say $f=gh$, and $\alpha$ is a root of $f$, then $\alpha$ must also be a root of one of the factors (say $g$), which is where the restrictions come from: all polynomial relationships have to be preserved by $\mathbb Q$-automorphisms, not just an arbitrarily chosen one. So such an automorphism preserves both $f(\alpha)=0$ and $g(\alpha)=0$. So $\alpha$ has to be mapped to a root of both $f$ and $g$ (so essentially a root of $g$).

Question 4:

Essentially, the roots of the polynomial must not admit too many polynomial relationships between the roots (as in, one root is the image of another under a polynomial), because those would put the restrictions discussed above on the choice of permutations.

Question 5 and 6:

Why would the order be 12? If the minimal polynomial of $F/\mathbb Q$ is $(X-a_1)\dots(X-a_4)$, then the minimal polynomial of $F/\mathbb Q(a_1)$ is $(X-a_2)\dots(X-a_4)$, just drop the factor with $a_1$. The Galois group of this polynomial is at most $S_3$. In fact, taking only the subgroup of $S_4$ whose permutations fix $a_1$ is essentially the same as taking the group of permutations of only $a_2,\dots,a_4$. That's $S_3$, realized as a subgroup of $S_4$. And this realization of $S_3$ as a subgroup contains a transposition.

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I'm going to comment only on Question 4, since other questions (as well as Q4) have been addressed in other answers.

Assume that $f$ is irreducible. What must be unique about the situation in order for $G$ to really be all of $S_n$?

Let $d$ be a fixed positive integer. Almost all polynomials over $\mathbb Q$ of degree at most $d$ will have $S_d$ as their Galois group, in the following sense: among all the polynomials of degree at most $d$ with absolute values of their coefficients bounded by $N>0$, the probability that a randomly chosen polynomial will have $S_d$ as its Galois group is $1$ as $N$ goes to infintiy.

So in this sense, nothing is unique in the situation where $G$ is all of $S_d$, that's actually the "generic" case. Those polynomials that don't have $S_d$ as their Galois group are the special ones.

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  • $\begingroup$ The set of $f \in \mathbb{Q}[x]$ with coeffs bounded by $N$ and of degree $\leq d$ is still not finite so you have no canonical probability there. You need to clear denominators before you start so they live in $\mathbb{Z}[x]$. $\endgroup$ Aug 8, 2020 at 22:54

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