Let $f\in\mathbb{Q}[x]$ be an irreducible polynomial of prime degree $p$, and exactly $2$ non-real roots.

You can view the Galois group of the splitting field as a subgroup of $S_p$. Complex conjugation shows that the Galois group contains a transposition. You can use Cauchy's theorem from group theory to show that the Galois group contains a $p$-cycle.

Then $f$ has Galois group $S_p$. This uses the slightly stronger fact that $S_p$ is generated by any transposition and $p$-cycle (which can be proved from the fact you mentioned).

In turn, constructing a polynomial with Galois group $S_5$ is useful for proving insolvability of the quintic.

Let $f\in\mathbb{Q}[x]$ be an irreducible polynomial of prime degree $p$, with exactly $2$ non-real roots.

You can view the Galois group of $f$ (i.e., the Galois group of the splitting $f$) as a subgroup of $S_p$. Complex conjugation shows that the Galois group contains a transposition. You can use Cauchy's theorem from group theory to show that the Galois group contains a $p$-cycle.

Then $f$ has Galois group $S_p$. This uses the slightly stronger fact that $S_p$ is generated by any transposition and $p$-cycle (which can be proved from the standard two-generating set).

In turn, constructing a polynomial with Galois group $S_5$ is useful for proving insolvability of the quintic.

Let $f\in\mathbb{Q}[x]$ be an irreducible polynomial of prime degree $p$, and exactly $2$ non-real roots.

You can view the Galois group of the splitting field as a subgroup of $S_p$. Complex conjugation shows that the Galois group contains a transposition. You can use Cauchy's theorem to show that the Galois group contains a $p$-cycle.

Then $f$ has Galois group $S_p$. This uses the slightly stronger fact that $S_p$ is generated by any transposition and $p$-cycle (which can be proved from the fact you mentioned).

In turn, constructing a polynomial with Galois group $S_5$ is useful for proving insolvability of the quintic.

Let $f\in\mathbb{Q}[x]$ be an irreducible polynomial of prime degree $p$, and exactly $2$ non-real roots.

You can view the Galois group of the splitting field as a subgroup of $S_p$. Complex conjugation shows that the Galois group contains a transposition. You can use Cauchy's theorem from group theory to show that the Galois group contains a $p$-cycle.

Then $f$ has Galois group $S_p$. This uses the slightly stronger fact that $S_p$ is generated by any transposition and $p$-cycle (which can be proved from the fact you mentioned).

In turn, constructing a polynomial with Galois group $S_5$ is useful for proving insolvability of the quintic.