What could be a general method to determine a $\mathbb{K}$-basis , example on $\mathbb{Q}(\sqrt{2},\sqrt{3})$

Question

What could be a general method to determine a $\mathbb{K}$-basis ,

For example , what would be a good strategy to determine a $\mathbb{Q}$-basis of $\mathbb{Q}(\sqrt{2},\sqrt{3})$ ?

I could find some answers on internet , but what i can't figure out is what basic "properties" or "theorems" are used along the way.

After a while , stackechange proposed me the article :

Basis for $\mathbb Q (\sqrt2 , \sqrt3 )$ over $\mathbb Q$

Edit : Something that could help me to understand : What is exactly the link between a minimal polynomial and finding the basis ?

Answer 1

First find a basis of $Q(\sqrt 2)$ over $Q$ $(1, \sqrt 2)$ then a basis of $Q(\sqrt 2, \sqrt 3)$ over $Q(\sqrt 2)$, say $(1, \sqrt 3)$ and multiply. here $1, \sqrt 2, \sqrt 3, \sqrt 6$

The general (easy) theorem is if $e_i, 1\leq i\leq n$ is a base of $L$ over $K$ and $f_j, 1\leq j \leq m$ a base of $M$ over $L$, then $e_i f_j$ is a base of $M$ over $K$.