*Goal*-- Create extensions from other extensions using operations such as intersection, compositum and listing subfields. They should be provided with appropriate coercion maps*Type*-- advanced features*Priority*-- High*Difficulty*-- High*Prerequisites*-- general extensions of rings, polynomial factoring*Background*--*Contributors*-- David Roe*Progress*- not started*Related Tickets*--

## Discussion

## Tasks

- Implement the compositum of two fields by factoring the defining polynomial for one over the other (I assume there's a better way...). Each of the original fields should coerce into the compositum.
- Implement the intersection of extension rings. Determine the precision of the defining polynomial... The intersection should coerce into each of the original fields.
- Implement an algorithm to give the lattice (in the poset sense) of subfields of a given p-adic field. These subfields should coerce naturally into the larger field.