Fix __getitem__ and slicing for p-adics

[saraedum]

This is a meta-ticket to track progress on fixing __getitem__ and slicing of p-adic elements and their .expansion().

Depends on #13299
Depends on #14304
Depends on #26406
Depends on #26407
Depends on #26408

Component: padics

Author: Julian Rueth

Issue created by migration from https://trac.sagemath.org/ticket/13300

[saraedum]

comment:1

The attached file (which has an extra 3 in the filename, sorry) fixes this.

To be consistent, indexing gives elements in the residue field. Mathematically, this seems to make sense, but is this really what one wants?

I'm running all doctests now to see if this breaks anything.

[saraedum]

comment:2

The doctests worked, so I'm setting this to needs review.

[saraedum]

Author: Julian Rueth

[roed314]

comment:4

Attachment: trac_13330.patch.gz

I don't think that giving elements of the residue field is what we want, though I agree that the current behavior for unramified extensions is silly. I would like to be able to have

sage: K.<a> = Qq(7^2, 20)
sage: z = K.random_element()
sage: z == sum([z[i] * 7^i for i in range(20)])
True

without having to lift from the residue field.

Of course, sometimes you want to get at the coefficient of a in z, or the coefficient of a in z[0]. Here's an idea. For matrices, we have the following::

sage: M = matrix(ZZ,4,4,range(16))
sage: M[0]
(0, 1, 2, 3)
sage: M[0,0]
0
sage: M[0,:]
[0 1 2 3]
sage: M[:,0]
[ 0]
[ 4]
[ 8]
[12]
sage: M[1:3,1:4]
[ 5  6  7]
[ 9 10 11]

What if we used a similar tuple indexing scheme for p-adic elements? For example:

sage: K.<a> = Qq(5^4,5)
sage: z = K.random_element(); z
a^3*5^-2 + (3*a^3 + 3*a^2)*5^-1 + (3*a^3 + a^2 + 3*a) + 4*a^3*5 + (4*a + 2)*5^2 + O(5^3)
sage: z[0]
3*a^3 + a^2 + 3*a + O(5^3)
sage: z[2]
4*a + 2 + O(5)
sage: z[2].parent()
Unramified Extension of 5-adic Field with capped relative precision 5 in a defined by (1 + O(5^5))*x^4 + (O(5^5))*x^3 + (4 + O(5^5))*x^2 + (4 + O(5^5))*x + (2 + O(5^5))
sage: z[0,0]
O(5^3)
sage: z[0,0].parent()
5-adic Field with capped relative precision 5
sage: z[2,0]
2 + O(5)
sage: z[2,1]
4 + O(5)
sage: z[:-1]
a^3*5^-2 + O(5^-1)
sage: z[:1]
a^3*5^-2 + (3*a^3 + 3*a^2)*5^-1 + (3*a^3 + a^2 + 3*a) + O(5)
sage: z[:] == z[:,:] == z
True
sage: all([z[i,:] == z[i] for i in range(-3,2)])
True
sage: z[:,0]
2*5^2 + O(5^3)
sage: z[:,0].parent()
5-adic Field with capped relative precision 5
sage: z[:,1]
3 + 4*5^2 + O(5^3)
sage: z[:,2]
3*5^-1 + 1 + O(5^3)
sage: z[:,3]
5^-2 + 3*5^-1 + 3 + 4*5 + O(5^3)
sage: sum([a^j * z[:,j] for j in range(4)]) == z
True
sage: z[-1:2,1:3]
3*a^2*5^-1 + (a^2 + 3*a) + O(5^2)

The parent of z[i], z[i,T] and z[S,T] (for integer i and slices S,T) would be the unramified extension; the parent of z[i,j] and z[S,j] would be the ground field (or ring). Negative indices would never be considered as counting backward from a length, even for ring elements.

What do you think?

[saraedum]

comment:8

Let's think about this again once #14304 has been merged.

[saraedum]

Changed dependencies from #13299 to #13299, #14304

[saraedum]

comment:10

I like your proposal. Let's discuss this at one of the IRC sessions.

[saraedum]

Changed dependencies from #13299, #14304 to #13299, #14304, #26406, 26407

[saraedum]

Changed dependencies from #13299, #14304, #26406, 26407 to #13299, #14304, #26406, #26407, #26408