Fix extend handling in FiniteField_givaroElement.sqrt

[bhuvan-somisetty]

Description

Fixes #42719

In src/sage/rings/finite_rings/element_givaro.pyx, FiniteField_givaroElement.sqrt(extend=True, all=True) previously returned [] (empty list) on non-squares instead of raising NotImplementedError.

This occurred because the if all: check returned [] immediately when self.is_square() was False, completely bypassing the elif extend: raise NotImplementedError branch. As a result, extend=True was silently ignored whenever all=True, falsely claiming the element had no square roots in any extension field and violating the API contract.

Proposed Solution

1. In FiniteField_givaroElement.sqrt(), check if extend: raise NotImplementedError upfront before checking all or evaluating base-field squareness (matching element_pari_ffelt.pyx and element_base.pyx).
2. Correct the docstring default parameter note for extend from (default: True) to (default: False) to match the method signature.
3. Add regression doctests for all combinations of (extend, all) for both squares and non-squares.

Checklist

• The title is concise and informative.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation and checked the documentation preview.

Dependencies

None.

Signed-off-by: bhuvan-somisetty somisettybhuvan5@gmail.com

[cxzhong]

please rebase to the lastest develop

[cxzhong]

Thanks for fixing the non-square all=True case from #42719. I think the new guard is too broad, though:

if extend:
    raise NotImplementedError

This also rejects elements whose roots already lie in the base field. For example, before this change, GF(9).one().sqrt(extend=True) returns 1, and GF(9).one().sqrt(extend=True, all=True) returns [1, 2]; after this change both raise NotImplementedError. The method documents extend=True as allowing an extension if necessary, so square elements should not require extension support.

Could you keep the existing elif extend: in the single-root path and add the exception only to the non-square all=True path?

if all:
    if self.is_square():
        a = self.sqrt()
        return [a, -a] if -a != a else [a]
    if extend:
        raise NotImplementedError
    return []

Please also add explicit extend=True tests for a square element, and ideally for characteristic 2, where every element is a square.

[bhuvan-somisetty]

Good catch, thank you! I've updated the logic so extend=True is allowed on elements whose roots already lie in the base field, and only raises NotImplementedError on non-squares. I also added doctests for square elements with extend=True as well as characteristic 2 fields.