Field extension as vector spaces

Question

Let $F$ be a field and $K/F$ a field extension. I have seen that we can consider the extension as a $F$ -vector space(scalars over $F$) Is this a kind of convention? If not ,is there a proof about this?

Answer 1

Let $$\star :K\times K\to K,(x,y)\mapsto x+y$$ where $+$ is the addition in $K$;

and$$\bullet :F\times K\to K, (k,x)\mapsto kx$$ where $kx$ is the result of the multiplication of $k$ and $x$ in $K(^1)$.

We can therefore use the usual notations for $\star$, the law of internal composition and $\bullet$, the law of external composition and verify for example that $(K,\star)$ is a commutative group as follows :

• $\forall x\in K, x+0=0+x=x$, where $0$ is the one of $K$;
• $\forall x,y \in K, x+(y+z)=(x+y)+z$, since it is true in $K$;
• $\forall x \in K, x+(-x)=(-x)+x=0$;
• $\forall x,y \in K, x+y=y+x$, since it is true in $K$.

(We could have gone faster by writing that $\star=+$ so that $(K,\star)$ is a commutative group since $(K,+) $ is)

The rest of the proof is just as easy...

And we can conclude $$(K,\star,\bullet)\text{ is a vector space over }F.\square$$

---

$(^1)$Recall that saying that K is an extension of F means that F is a subfield of K. So $$k\in F\implies k\in K$$