Remarks on certain composita of fields

TL;DR

This paper investigates the algebraic closure properties of composita of algebraically closed fields, showing they are not necessarily algebraically closed and analyzing their Galois groups, with new results on field composita.

Contribution

It proves that composita of algebraically closed fields are not always algebraically closed when linearly disjoint over their intersection, and characterizes their Galois groups, providing new insights into field composita.

Findings

Compositum of algebraically closed fields is not necessarily algebraically closed.

Galois group of maximal abelian extension of compositum is free pro-abelian.

Free pro-nilpotent group of rank equal to the intersection's size can be realized as a Galois group.

Abstract

Let $L$ and $M$ be two algebraically closed fields contained in some common larger field. It is obvious that the intersection $C=L\cap M$ is also algebraically closed. Although the compositum $LM$ is obviously perfect, there is no reason why it should be algebraically closed except when one of the two fields is contained in the other. We prove that if the two fields are strictly larger that $C$, and linearly disjoint over $C$, then the compositum $LM$ is not algebraically closed; in fact we shall prove that the Galois group of the maximal abelian extension of $LM$ is the free pro-abelian group of rank $|LM|$, and that the free pro-nilpotent group of rank $|C|$ can be realized as a Galois group over $LM$. The above results may be considered as the main contribution of this article but we obtain some additional results on field composita that might be of independent interest.