Add inverse_of_unit() for padics

[saraedum]

Currently, the padics lack an inverse_of_unit() method:

sage: (-1).inverse_of_unit()
-1
sage: ZpCA(3)(-1).inverse_of_unit()
AttributeError

The drawback with simply using ~ is that it puts the result in the fraction field and it can be annoying to always convert it back to the original ring when implementing general algorithms for all padics rings and fields:

sage: t = ZpCA(3,2)(-1)
sage: t
2 + 2*3 + O(3^2)
sage: t.parent()
3-adic Ring with capped absolute precision 2
sage: ~t
2 + 2*3 + O(3^2)
sage: (~t).parent()
3-adic Field with capped relative precision 2

The attached patch implements a method inverse_of_unit() and fixes a conversion error that came up when testing it.

Depends on #13600

Component: padics

Author: Julian Rueth

Reviewer: David Roe

Merged: sage-5.8.beta2

Issue created by migration from https://trac.sagemath.org/ticket/13539

[saraedum]

Description changed:

--- 
+++ 
@@ -12,9 +12,13 @@
 ```
 sage: t = ZpCA(3,2)(-1)
 sage: t
+2 + 2*3 + O(3^2)
 sage: t.parent()
+3-adic Ring with capped absolute precision 2
 sage: ~t
+2 + 2*3 + O(3^2)
 sage: (~t).parent()
+3-adic Field with capped relative precision 2
 ```
 
 The attached patch implements a method `inverse_of_unit()` and fixes a conversion error that came up when testing it.

[roed314]

comment:3

I'm happy with the changes to LocalGenericElement. There are changes to pAdicZZpXCAElement though that I think make the code less readable (and actually change the functionality a bit). Were those intended?

[saraedum]

Dependencies: #13600

[saraedum]

comment:4

You're right. I didn't understand these changes myself anymore now. I tried to put them into a separate ticket and turned them into something that is hopefully more readable #13600.

[roed314]

comment:5

Cool. Once you update the patch here and it passes doctests I can give it a positive review

[saraedum]

comment:6

I removed the pieces that are now in #13600. I also added a copyright notice since it was missing (I took the years from the repo history).