Solution
= Solution
Over $\mathbb F_3$,
$$
t^4+1=(t^2+t+2)(t^2+2t+2).
$$
Both quadratics have <discriminant> $2$, a nonsquare in $\mathbb F_3$, so are <irreducible polynomials>. Both split in the same quadratic <finite field> $\mathbb F_9$. Consequently $\boxed{\operatorname{Gal}(t^4+1/\mathbb F_3)\cong C_2}$. Its generator is the <finite-field Frobenius automorphism> $x\mapsto x^3$, acting as a product of two transpositions on the four roots.