Remarks on certain composita of fields

Details Remarks on certain composita of fields Remarks on certain composita of fields

2012-01-19

arxiv.org/abs/1201.4105v1

Mathematics

Commutative Algebra

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Remarks on certain composita of fields does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Remarks on certain composita of fields, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Remarks on certain composita of fields will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-254512