Although the result is intuitive, I'm still not certain about inverse elements in $L$. For instance, if $K=\mathbb Q$ and $L=\mathbb Q(\sqrt2,\sqrt3,\sqrt5,\sqrt7)$, then why is
$$\frac{1}{\sqrt2+\sqrt3+\sqrt5+\sqrt7}$$
a linear combination of $\{1,\sqrt 2, \sqrt 3, \sqrt 6, \dots, \sqrt{210}\}$?
Now, actually, I'm aware of the fact that in the case of extensions of $\mathbb Q$ with surds, we can compute the inverse: $$-185 \sqrt{2}+145 \sqrt{3}+133 \sqrt{5}-135 \sqrt{7}-62 \sqrt{30}+50 \sqrt{42}+34 \sqrt{70}-22\sqrt{105},$$
but this was painstaking to compute, is there an easy way to see why this holds for any field extension $L/K$?
Best Answer
The additive law is the additive law of $L$ (and $K$) . The external law is given by multiplication in $L$: $(\lambda, x)\in K\times L\mapsto \lambda x\in L$. It is easy to check that these two laws verify the axioms of a $K$-vector space.
Note that the extension $L/K$ does not need to be algebraic (you could take $L=K(X),$ where $X$ is an inderminate).
The fact that in your example $\frac{1}{\sqrt2+\sqrt3+\sqrt5+\sqrt7}$ is a linear combination of $\{1,\sqrt 2, \sqrt 3, \sqrt 6, \dots, \sqrt{210}\}$ has less to do with the fact that $L$ is a $K$-vector space, but more to do with the fact that $L/K$ is algebraic, which is more a ring theoritical reason.
More precisely if $\alpha_1,\ldots,\alpha_n$ are algebraic over $K$, then $K(\alpha_1,\ldots,\alpha_n)=K[\alpha_1,\ldots,\alpha_n]$. This comes from the case $n=1$ by induction.