Galois group of $x^3 - 2 $ over $\mathbb Q$

Question

I know the Galois group is $S_3$. And obviously we can swap the imaginary cube roots. I just can't figure out a convincing, "constructive" argument to show that I can swap the "real" cube root with one of the imaginary cube roots.

I know that if you have a 3-cycle and a 2-cycle operating on three elements, you get $S_3$. I have a general idea that based on the order of the group there's supposed to be at least a 3-cycle. But this doesn't feel very "constructive" to me.

I wonder if I've made myself understood in terms of what kind of argument I'd like to see?

Answer 1

A brute-force way to see it is easy enough. The roots of $X^3 - 2$ are $\sqrt[3]{2},\; \omega\sqrt[3]{2},\; \omega^2\sqrt[3]{2}$, so the splitting field of your polynomial is $K = \mathbb{Q}(\sqrt[3]{2},\omega)$. You are asking why it's legitimate to send $\sqrt[3]{2}$ to $\omega\sqrt[3]{2}$. Well, let's try it. Let $\sigma: K \to K$ be a map that sends $\sqrt[3]{2}$ to $\omega\sqrt[3]{2}$. How are we going to extend this to an element of the Galois group? We need $\sigma$ to be an automorphism that fixes $\mathbb{Q}$.

Our field $K$ is a $\mathbb{Q}$-vector space generated by six basis elements: $1,\; \sqrt[3]{2},\; \sqrt[3]{4},\; \omega,\; \omega\sqrt[3]{2},\; \omega\sqrt[3]{4}$. If we can define how $\sigma$ acts on those, we can extend linearly to the whole space: that is, we can extend $\sigma$ such that $\sigma(a+b) = \sigma(a) + \sigma(b)$ for all $a$ and $b$. We also know that, if we define $\sigma$ sensibly, we should get that $\sigma(ab) = \sigma(a) \sigma(b)$. By "sensibly" here, I mean that $\sigma$ should send each element of $K$ to one of its conjugates - that is, it should send $x$ to any $y$ that satisfies the same minimal polynomial as $x$. We also know that it's enough to make sure $\sigma$ is multiplicative on the basis, and in fact we only need to define it on $\sqrt[3]{2}$ and $\omega$, since the rest will then follow automatically by multiplicativity.

So we have a few options from what we've deduced so far. $\sigma$ can send $\sqrt[3]{2}$ to any of $\sqrt[3]{2},\; \omega\sqrt[3]{2},\; \omega^2\sqrt[3]{2}$, and it can send $\omega$ to either of $\omega$ and $\omega^2$. So let's say $\sigma(\sqrt[3]{2}) = \omega\sqrt[3]{2}$ (as we wanted) and $\sigma(\omega) = \omega$ (no good reason for this choice, but we had to make one). Where do all the other basis elements go? Can you convince yourself now that $\sigma$ is an element of the Galois group?

(What would have happened if we'd chosen $\sigma(\omega) = \omega^2$?)

Answer 2

What you're probably missing is that, although the real cube root of 2 stands out, all three complex cube roots of 2 are algebraically the same, that is, they cannot be told apart, and so can be permuted at will. Hence $S_3$.

Contrast this with the seemingly analogous situation of, say, the 6-th roots of unity. Here the roots are not the same algebraically, because $x^6-1$ is not irreducible.

Answer 3

I don't know what kind of argument you'd like to see, but the standard way to compute the Galois group of an irreducible cubic (characteristic not $2$) is to look at the sign of its discriminant $\Delta = \prod_{i \neq j} (r_i - r_j)$. It has a square root $\prod_{i < j} (r_i - r_j)$ in a splitting field of the polynomial which is multiplied by $1$ when an even permutation is applied to the roots and multiplied by $-1$ when an odd permutation is applied to the roots.

By the fundamental theorem of Galois theory, it follows that the Galois group of an irreducible polynomial of degree $n$ is contained in $A_n$ if and only if the discriminant is a square. In your example, the discriminant of the cubic $x^3 + px + q$ is given by $-4p^3 - 27q^2 = -108$ which is not a square, so the Galois group is not contained in $A_3 \cong C_3$, hence must be $S_3$.

Another way is to use Dedekind's theorem that if an irreducible polynomial $f$ factors into distinct irreducible factors $f = \prod f_i \bmod p$, then the Galois group of $f$ has a permutation with cycle type $(\deg f_1, \deg f_2, ...)$. Thus to show that there is a $3$-cycle in the Galois group of $x^3 - 2$ it suffices to find a prime with respect to which this polynomial is irreducible, and this is easy since for cubics irreducibility is equivalent to not having a root. $p = 7$ works.

Answer 4

The question implicitly assumes the base field is $Q$. The Galois group of a polynomial only exists with reference to a specified field containing the coefficients. It is not enough to assume that $X^3 - 2$ is irreducible, because the Galois group is cyclic of order 3 (that is, a proper subgroup of S_3 rather than the whole thing) if the base field contains primitive cube roots of 1, and $S_3$ otherwise.

If the base field contains no cube roots of 1 or 2, then a splitting field of $X^3 -2$ is obtained by adjoining all the cube roots of 1 and 2. If the adjoined roots are taken from a larger "pre-existing" field such as the complex numbers, an abstract choice of splitting field is equivalent to selecting one nontrivial cube root of 1 (two choices) and one cube root of 2 (three choices). There are six possibilities and they differ by elements of $S_3$.

The group structure is not the direct product of the cyclic subgroups of order 2 and 3, because the operation of changing one's choice of $\sqrt[3]{1}$ will also re-arrange the cube roots of 2, exchanging the two unselected $\sqrt[3]{2}$'s.

Answer 5

One can prove a more general result : Let $f(X)$ be an irreducible polynomial with rational coefficients and of degree $p$ prime. If $f$ has precisely two nonreal roots in the complex numbers, then the Galois group of $f$ is $S_p$

So taking here $p=3$ we're done

Answer 6

I recently found an answer to my own question that no one else has brought up, so I thought I'd post it. The thing that always bothered me about swapping the real and imaginary cube roots of two was the asymmetry of it all. What makes the positive imaginary root distinguishable from the negative imaginary root?

For me at least, the asymmetry goes away when you realize that the same permutation that exchanges the real cube root of two with the positive imaginary root ALSO exchanges the real cube root of FOUR with the negative imaginary root.

Answer 7

Let me try something here.

It's well-known that the Galois group of $\Bbb Q(\zeta_n)/\Bbb Q$ is $\Bbb Z_n^×.$ So when $n$ is prime the extension is degree $p-1$ (and btw the Galois group is cyclic).

So here we have $\zeta_3$ and the extension $\Bbb Q(\zeta_3) $ of $\Bbb Q$ is degree $3-1=2$.

Then note that $\Bbb Q(\sqrt[3]{2},\zeta_3)/\Bbb Q(\zeta_3) $ is an extension of degree $3.$

Thus we have $3\cdot 2=6$ for the degree of our extension.

Since you can check the Galois group is non-abelian, we get $S_3.$

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$x^p-2$ is just the same, giving us an extension of degree $p(p-1).$

For different $p$ however some more fancy footwork is required to determine the Galois groups (that the extensions are normal and separable follows from some standard arguments). For instance, for $p=5$ we get a group of order $20.$ But which one?

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One candidate is the Frobenius group, which in this case is also the holomorph, $\Bbb Z_{p-1}\ltimes \Bbb Z_p.$

And a little checking will show that this is it.

Since $\Bbb Z_p$ can easily be verified to be normal (via Sylow, say) inside $G,$ and $\Bbb Z_p\Bbb Z_{p-1}=G,$ we do indeed have a non-trivial semi-direct product.