Galois Group of splitting field of $ X^5 - 4X + 6 $ over $\mathbb{Q}$

Question

I need help finding the Galois group of the next polynomial:

$ X^5 - 4X + 6 $

So far I know that by Eisenstein's criterion, for the prime 2, the polynomial is irreducible.

Answer 1

First, the polynomial is irreducible by an application of Eisenstein's criterion for $p=2$.

Next we use Dedekind's theorem, which states that when given a monic irreducible polynomial $f(X)\in\mathbb{Z}[X]$ and a prime $p$ such that $f\pmod{p}$ has simple roots, then $G_f$ contains an automorphism that permutes the roots of $f$ according to a permutation with cycle type $(\deg(g_1),\deg(g_2), \dots , \deg(g_k))$, where $f$ decomposes into irreducibles mod $p$ as $f(X)\equiv g_1(X)\dots g_{k}(X) \pmod{p}$ (with all $g_i$'s not necessarily distinct).

To apply the above fact we factor $X^5-4X+6$ modulo $p$ into irreducibles for some small primes:

$X^5-4X+6\equiv X^5\pmod{2} \\X^5-4X+6\equiv X(X+1)(X+2)(X^2+1)\pmod{3}\\X^5-4X+6\equiv (X+1)(X^2+X+1)(X^2+X+2)\pmod{5}\\X^5-4X+6\equiv X^5-4X+6\pmod{7}\\\vdots$

Looking at $p=3$ we find a cycle of type (1,1,1,2), that is, a transposition. At $p=7$, $f\pmod{7}$ is irreducible, so $G_f$ contains a 5-cycle.

Any 5-cycle and transposition generate $S_5$, so we find that $G_f=S_5$ as $G_f$ is contained in $S_5$.

Personally I would recommend these notes (or others) by Keith Conrad for more examples.