*Goal*-- Design a specialized p-adic extension type for cyclotomic extensions of Qp and Zp. Design element classes for unramified extensions that take advantage of a Gauss normal basis for faster arithmetic.*Type*-- speed improvements, coherence with number fields*Priority*-- Medium*Difficulty*-- Medium-Hard*Prerequisites*-- Cyclotomic fields in general will need to wait on polynomial factoring*Background*-- Look at Lercier's talk from Counting Points: Theory, Algorithms and Practice*Contributors*-- David Roe, David Lubicz*Progress*- not started*Related Tickets*--

## Discussion

## Tasks

- Implement Gauss normal basis for finite fields.
- Implement Gauss normal basis for unramified extensions of Zp and Qp.
- Implement elliptic normal basis for finite fields.
- Implement elliptic normal basis for unramified extensions of Zp and Qp.
Implement a special parent for cyclotomic extensions. For totally ramified and unramified cyclotomic extensions this can be done now; in general it will need to wait on polynomial factoring.