Relating the $F$-rational points of a torus to the character group If $T$ is an algebraic torus over a field $F$, then I keep reading that $T(F) \cong X_*(T) \otimes F^\times$, where $X_*(T)$ is the cocharacter lattice. What is the isomorphism between them?

This is not true unless $T$ is split. If $T$ is split the isomorphism is
$$X_\ast(T)\otimes F^\times\to T(F), \qquad \alpha\otimes c\mapsto \alpha(c).$$
So, for intance if $T=\mathbb{G}_{m,F}^n$ then
$$X_\ast(T)=\left\{\sum_i a_i e_i: a_i\in\mathbb{Z}\right\}$$
where $$e_i:\mathbb{G}_{m,F}\to \mathbb{G}_{m,F}^n,\qquad a\mapsto (1,\ldots,a,\ldots,1).$$
Our map then takes
$$\sum_i e_i \otimes c_i\mapsto (c_1,\ldots,c_n)$$
where now here each $c_i$ is allowed to be in $F^\times$.
