> A torus (sometimes called an algebraic torus) over a field is an algebraic group that can be described as a direct product of finitely many multiplicative groups of finite extensions of that field. Note that we could use the multiplicative group of the field itself in one or more of the direct factors.

The term split torus is used for a torus that is a direct product of copies of the multiplicative group of the field. When the field is an algebraically closed field, any torus over it is a split torus.

@Daminark @LeakyNun generally if $\operatorname{Gal}(E/F)$ is cyclic of order $n$ then $E/F$ is the splitting field of some irreducible polynomial $f$ of degree $n$.
The reason is that the embedding into $S_{\operatorname{deg}(f)}$ given by the action on the roots must be generated by a single $n$-cycle (since having multiple cycles would give different orbits and a non-transitive action) and the only way the subgroup generated by a cycle acts transitively is if the length of the cycle is equal to the number of elements it acts on

@LeakyNun there's another way to define group schemes (affine or not) which I actually prefer. So basically by a Yoneda argument, we can view any scheme over $k$ as a functor $F:k-\mathbf{Alg} \to \mathbf{Set}$ by looking at some Hom-functor.
One can show, by a Yoneda exercise I won't spoil for you that having a group object is equivalent to having a functor $\tilde{F}: k-\mathbf{Alg} \to \mathbf{Grp}$ such that $\tilde{F}$ composed with the forgetful functor $\mathbf{Grp} \to \mathbf{Set}$ is $F$