Let $F/K$ be a field extension and $L_1,L_2$ be subfields of $F$ containing $K$.
There is a notation I don't know what it's called, it's $L_1L_2$, defined by the subfield generated by $L_1 \cup L_2$. So can anyone give me the name?
Furthermore, do we have an explicit describe for this notation? I mean suppose $L_1,L_2$ are vector spaces over K with finitely dimensions, can we find the basis of $L_1L_2$ or something like that? Thank you
It's called the compositum of the two field extensions. if $ L_1/K $ and $ L_2/K $ are both finite, then $ L_1 L_2 / K $ in a larger field $ F $ is spanned by elements of the form $ \alpha_i \beta_j $, where $ \alpha_i $ is a basis for $ L_1/K $ and $ \beta_j $ is a basis for $ L_2/K $. These elements may fail to be linearly independent, so they need not form a basis for $ L_1 L_2/K $, but one may always compute a basis by using row reduction over $ K $ upon writing the $ \alpha_i \beta_j $ in a chosen basis for $ F/K $.