The names of $\Bbb{G}_a$ and $\Bbb{G}_m$

Question

By wikipedia, over a field $k$, $\Bbb{G}_a=k$ and $\Bbb{G}_m=k^*$ as scheme over $k$. My question is why these group schemes calles "the additive group scheme" and "the multiplicative group scheme" respectively.

Answer 1

I will illustrate the multiplicative group as example, the additive group being similar.

The scheme $\Bbb G_m$ has the following property: given any field extension $K/k$, we have a natural bijection between the set of $K$-rational points $\Bbb G_m(K)$ and $K^*$, the multiplicative group of the field $K$. Moreover, the group structure of $\Bbb G_m$ translates exactly to the group structure of $K^*$.

In other words, taking $K$-rational points of $\Bbb G_m$ "produces" the multiplicative group of the field $K$: $$\Bbb G_m(K)``="\text{multiplicative group of }K.$$ It therefore makes sense to call $\Bbb G_m$ "the multiplicative group scheme".