Is $x^6 + 108$ irreducible over $\mathbb{Q}$?

Question

I'm trying to determine whether or not $x^6 + 108$ is irreducible over $\mathbb{Q}$. Is there an easy way to determine this ? I tried Eisenstein's Criterion directly, and with the substitutions $x \longmapsto x + 1, x \longmapsto x-1, x \longmapsto x+2, x \longmapsto x-2$ to try and show it is irreducible, but that didn't work out. Clearly, there's no linear factor of $x^6 + 108$ in $\mathbb{Q}$, since $x^6 + 108$ doesn't have a root in $\mathbb{Q}$. Showing there's no two cubic polynomials in $\mathbb{Q}[x]$ that multiply to $x^6 + 108$, or no cubic and quadratic polynomial in $\mathbb{Q}[x]$ that multiply to $x^6 + 108$ also quickly becomes a hassle. Is there an easier way to determine this more quickly?

Thanks!

Answer 1

Hint Since the polynomial $$x^6 + 108$$ is monic, if the field factors over $\Bbb Q$, it factors over $\Bbb Z$. Reducing modulo $7$, we have $$x^6 + 108 \pmod 7 \equiv x^6 - 4 \pmod 7 \equiv (x^3 + 2) (x^3 - 2) \pmod 7 .$$

Thus, if $x^6 + 108$ factors over $\Bbb Q$, it factors as a product of two cubics.

Answer 2

Let $x^6 + 108 = p(x)q(x)$ for non-constant polynomials $p,q$ with coefficients in $\mathbb Z$. Then, note that $p,q$ cannot have any real roots, since $x^6 + 108$ does not. This rules out any of $p,q$ having odd degree. Thus, it must happen that $\deg p = 2$ and $\deg q = 4$, without loss of generality.

Therefore, going modulo $p$ for a prime $p$ we get $x^6 +108 = \overline{p(x)}\overline{q(x)}$ modulo $p$. However, note that $\overline{p}$ and $\overline{q}$ must be of degree at most $2$ and at most $4$, since the degree cannot increase while going modulo $p$.

In particular, suppose that $x^6 + 108 = p_1(x)p_2(x)$ for some cubic $p_1,p_2$ which are irreducible modulo $p$. Then, we have $\bar p \bar q = p_1p_2$ as two distinct factorizations of $x^6 + 108$. By irreducibility in $\mathbb Z/p \mathbb Z$ we get that $\bar p$ is either a multiple of either $p_1$ or $p_2$, a contradiction by degree.

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Now, we look at $x^6 + 108$ modulo $7$. This gives $x^6-4$ modulo $7$, which becomes $(x^3 + 2)(x^3 - 2)$ , both of which are irreducible modulo $7$ since the only cubic residues modulo $7$ are $0,\pm 1$. Thus, the statement follows.

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Also, I am inclined to think that $x^6 + 108$ is in fact reducible mod $\mathbb Z/p \mathbb Z$ for every $p$, but I can't see it immediately.

EDIT : As Jyrki points out below, the splitting field of the polynomial $x^6 + 108$ is the same as the splitting field of $x^3- 2$ (this is fairly easy to see from the factorization $108 =2^23^3$). Therefore the Galois group of $x^6 + 108$ is $S_3$, since that is true of $x^3 - 2$.

However, the brilliant Dedekind lemma, as stated by Yuan in the link provided in Jyrki's comment, has as a corollary the following : if $f$ is irreducible modulo $p$ for any $p$ not dividing the discriminant of $f$, then the Galois group of $f$ must contain an element of order $\deg f$.

One can calculuate the discriminant of $x^6 - 2$/ use other ways to see that it only has $2$ and $3$ as prime factors. Since the Galois group contains no elements of order $6$, it immediately follows that $x^6 + 108$ is reducible modulo $p$ for every $p > 3$. Along with reducibility for $p=2,3$ this proves the assertion I had made earlier.