Let $L$ be a field, and $\{ E_i \}_{i \in I}$ be a family of subfields of $L$, that is, $E_i \subset L$ for each $i \in I$. Then, we define the compositum of the $E_i$'s to be the smallest field contained in $L$ that contains each of the $E_i$'s. More precisely, we define the compositum of the $E_i$'s to be
$$
\bigcap_{\substack{E_i \subset K \subset L \\ \text{for all } i \in I}} K,
$$
and it is easy to show that this is a field which is contained in any subfield of $L$ that contains $E_i$ for all $i \in I$.
My question is whether there is any standard notation to denote the compositum of a family of fields as above. I already know the following:
- If $I = \{ 1, \dots, n \}$, then the compositum of $E_1,\dots,E_n$ is usually denoted by $$E_1 \cdots E_n.$$
- If each $E_i$ is an extension of a field $k$, then one can denote the compositum by $$k(S),\quad \text{where } S = \bigcup_{i \in I} E_i.$$
But, if I don't have such a $k$ and if $I$ is an arbitrary indexing set, then is there a standard notation for the compositum? I have been reading Serge Lang's Algebra (3rd edition) and Patrick Morandi's Field and Galois Theory and no notation for the compositum (in the general scenario) is given in them.
Best Answer
You can always choose a $k$ for your second notation: both the prime field of $L$ and the intersection of all the $E_i$ would work.