Zariski Closure in a Torus and the dimension of the quotient

Question

I'm reading the second chapter of Serre's Abelian l-Adic Representations and Elliptic Curves, and need help with the first exercise. Let $K$ be a number field and let $T$ be the Weil Restriction of the multiplicative group $\mathbb{G}_m$ over $K$, to the rationals, i.e. $T=R_{K/\mathbb{Q}}(\mathbb{G}_{m/K})$. Let $E$ be the group of units in the ring of integers of $K$, hence a subgroup of $K^*=T(\mathbb{Q})$. Denote $\bar{E}$ to be the Zariski closure of $E$ in $T$ and $T_E$ to be the torus $T/\bar{E}$. The exercise asks to show that the dimension of $T_E$ is 2 for imaginary quadratic fields and 1 for both real quadratic fields and cubic number fields with one complex place. I have just started reading about algebraic groups and am struggling with finding the Zariski closure of the unit groups in the tori. I know what the group $E$ looks like, with the help of Dirichlet's unit theorem, but am unable to proceed further. Any hints would be appreciated.