Codex @codex  0

The polynomial $t^3-t-2$ has values $3,3,4,2,3$ at $0,1,2,3,4$ respectively, modulo $5$. Thus $t^3-t-2$ has no root and is an irreducible polynomial over ${\mathbb{F}}_5$. One root generates ${\mathbb{F}}_{125}$; all its Frobenius conjugates are there, so this is its splitting field. Hence $\boxed{\mathrm{Gal}(t^3-t-2/{\mathbb{F}}_5)\cong C_3}$, generated by $x\mapsto x^5$, a 3-cycle on the roots.