gh-39302: adding the category keyword to padic rings and fields classes
    
in the hope that this will allow to simplify something later about
`is_field`..

### 📝 Checklist

- [x] 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.
    
URL: #39302
Reported by: Frédéric Chapoton
Reviewer(s): Kwankyu Lee

Author: Release Manager

src/sage/rings/padics/factory.py

@@ -17,7 +17,7 @@
     sage: R = QpLF(2)
 """
 
-#*****************************************************************************
+# ****************************************************************************
 #       Copyright (C) 2007-2013 David Roe <[email protected]>
 #                               William Stein <[email protected]>
 #
@@ -26,10 +26,11 @@
 #  the License, or (at your option) any later version.
 #
 #                  https://www.gnu.org/licenses/
-#*****************************************************************************
+# ****************************************************************************
 
 from sage.misc.superseded import experimental
 
+from sage.categories.fields import Fields
 from sage.structure.factory import UniqueFactory
 from sage.rings.integer import Integer
 from sage.rings.infinity import Infinity
@@ -783,35 +784,45 @@ def create_object(self, version, key):
                 pass
             p, prec, type, print_mode, name, print_pos, print_sep, print_alphabet, print_max_terms, show_prec, label = key
 
+        _Fields = Fields()
+
         if type == 'capped-rel':
             if print_mode == 'terse':
                 return pAdicFieldCappedRelative(p, prec, {'mode': print_mode, 'pos': print_pos, 'sep': print_sep, 'alphabet': print_alphabet,
-                                                          'ram_name': name, 'max_terse_terms': print_max_terms, 'show_prec': show_prec}, name)
+                                                          'ram_name': name, 'max_terse_terms': print_max_terms, 'show_prec': show_prec}, name,
+                                                category=_Fields)
             else:
                 return pAdicFieldCappedRelative(p, prec, {'mode': print_mode, 'pos': print_pos, 'sep': print_sep, 'alphabet': print_alphabet,
-                                                          'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name)
+                                                          'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name,
+                                                category=_Fields)
         elif type == 'floating-point':
             if print_mode == 'terse':
                 return pAdicFieldFloatingPoint(p, prec, {'mode': print_mode, 'pos': print_pos, 'sep': print_sep, 'alphabet': print_alphabet,
-                                                          'ram_name': name, 'max_terse_terms': print_max_terms, 'show_prec': show_prec}, name)
+                                                         'ram_name': name, 'max_terse_terms': print_max_terms, 'show_prec': show_prec}, name,
+                                               category=_Fields)
             else:
                 return pAdicFieldFloatingPoint(p, prec, {'mode': print_mode, 'pos': print_pos, 'sep': print_sep, 'alphabet': print_alphabet,
-                                                         'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name)
+                                                         'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name,
+                                               category=_Fields)
         elif type == 'relaxed':
             if print_mode == 'terse':
                 return pAdicFieldRelaxed(p, prec, {'mode': print_mode, 'pos': print_pos, 'sep': print_sep, 'alphabet': print_alphabet,
-                                                'ram_name': name, 'max_terse_terms': print_max_terms, 'show_prec': show_prec}, name)
+                                                   'ram_name': name, 'max_terse_terms': print_max_terms, 'show_prec': show_prec}, name,
+                                         category=_Fields)
             else:
                 return pAdicFieldRelaxed(p, prec, {'mode': print_mode, 'pos': print_pos, 'sep': print_sep, 'alphabet': print_alphabet,
-                                                'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name)
+                                                   'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name,
+                                         category=_Fields)
         elif type[:8] == 'lattice-':
             subtype = type[8:]
             if print_mode == 'terse':
                 return pAdicFieldLattice(p, prec, subtype, {'mode': print_mode, 'pos': print_pos, 'sep': print_sep, 'alphabet': print_alphabet,
-                                                            'ram_name': name, 'max_terse_terms': print_max_terms, 'show_prec': show_prec}, name, label)
+                                                            'ram_name': name, 'max_terse_terms': print_max_terms, 'show_prec': show_prec}, name, label,
+                                         category=_Fields)
             else:
                 return pAdicFieldLattice(p, prec, subtype, {'mode': print_mode, 'pos': print_pos, 'sep': print_sep, 'alphabet': print_alphabet,
-                                                            'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name, label)
+                                                            'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name, label,
+                                         category=_Fields)
         else:
             raise ValueError("unexpected type")
 

src/sage/rings/padics/generic_nodes.py

@@ -314,7 +314,7 @@ class pAdicLatticeGeneric(pAdicGeneric):
         sage: R._prec_type()
         'lattice-float'
     """
-    def __init__(self, p, prec, print_mode, names, label=None):
+    def __init__(self, p, prec, print_mode, names, label=None, category=None):
         """
         Initialization.
 
@@ -357,7 +357,7 @@ def __init__(self, p, prec, print_mode, names, label=None):
         else:
             raise ValueError("subtype must be either 'cap' or 'float'")
         self._element_class = self.__make_element_class__(element_class)
-        pAdicGeneric.__init__(self, self, p, prec, print_mode, names, None)
+        pAdicGeneric.__init__(self, self, p, prec, print_mode, names, None, category=category)
 
     def _prec_type(self):
         """

src/sage/rings/padics/padic_base_generic.py

@@ -34,7 +34,7 @@
 class pAdicBaseGeneric(pAdicGeneric):
     _implementation = 'GMP'
 
-    def __init__(self, p, prec, print_mode, names, element_class):
+    def __init__(self, p, prec, print_mode, names, element_class, category=None):
         """
         Initialization.
 
@@ -47,7 +47,7 @@ def __init__(self, p, prec, print_mode, names, element_class):
             self.prime_pow = PowComputer_flint(p, 1, 1, 1, self.is_field())
         else:
             self.prime_pow = PowComputer(p, max(min(prec - 1, 30), 1), prec, self.is_field(), self._prec_type())
-        pAdicGeneric.__init__(self, self, p, prec, print_mode, names, element_class)
+        pAdicGeneric.__init__(self, self, p, prec, print_mode, names, element_class, category=category)
         if self.is_field():
             if self.is_capped_relative():
                 coerce_list = [pAdicCoercion_ZZ_CR(self), pAdicCoercion_QQ_CR(self)]

src/sage/rings/padics/padic_base_leaves.py

@@ -211,7 +211,7 @@ class pAdicRingCappedRelative(pAdicRingBaseGeneric, pAdicCappedRelativeRingGener
     An implementation of the `p`-adic integers with capped relative
     precision.
     """
-    def __init__(self, p, prec, print_mode, names):
+    def __init__(self, p, prec, print_mode, names, category=None):
         """
         Initialization.
 
@@ -247,7 +247,7 @@ def __init__(self, p, prec, print_mode, names):
             ....:                  max_runs=2^5, skip='_test_log')
             sage: R._test_log(max_runs=2, elements=[R.random_element() for i in range(4)])          # long time
         """
-        pAdicRingBaseGeneric.__init__(self, p, prec, print_mode, names, pAdicCappedRelativeElement)
+        pAdicRingBaseGeneric.__init__(self, p, prec, print_mode, names, pAdicCappedRelativeElement, category=category)
 
     def _coerce_map_from_(self, R):
         """
@@ -309,7 +309,7 @@ class pAdicRingCappedAbsolute(pAdicRingBaseGeneric, pAdicCappedAbsoluteRingGener
     r"""
     An implementation of the `p`-adic integers with capped absolute precision.
     """
-    def __init__(self, p, prec, print_mode, names):
+    def __init__(self, p, prec, print_mode, names, category=None):
         """
         Initialization.
 
@@ -345,7 +345,7 @@ def __init__(self, p, prec, print_mode, names):
             ....:                  max_runs=2^5, skip='_test_log')
             sage: R._test_log(max_runs=2, elements=[R.random_element() for i in range(4)])
         """
-        pAdicRingBaseGeneric.__init__(self, p, prec, print_mode, names, pAdicCappedAbsoluteElement)
+        pAdicRingBaseGeneric.__init__(self, p, prec, print_mode, names, pAdicCappedAbsoluteElement, category=category)
 
     def _coerce_map_from_(self, R):
         """
@@ -410,7 +410,7 @@ class pAdicRingFloatingPoint(pAdicRingBaseGeneric, pAdicFloatingPointRingGeneric
     An implementation of the `p`-adic integers with floating point
     precision.
     """
-    def __init__(self, p, prec, print_mode, names):
+    def __init__(self, p, prec, print_mode, names, category=None):
         """
         Initialization.
 
@@ -446,7 +446,7 @@ def __init__(self, p, prec, print_mode, names):
             ....:                  max_runs=2^5, skip='_test_log')
             sage: R._test_log(max_runs=2, elements=[R.random_element() for i in range(4)])
         """
-        pAdicRingBaseGeneric.__init__(self, p, prec, print_mode, names, pAdicFloatingPointElement)
+        pAdicRingBaseGeneric.__init__(self, p, prec, print_mode, names, pAdicFloatingPointElement, category=category)
 
     def _coerce_map_from_(self, R):
         """
@@ -505,7 +505,7 @@ class pAdicRingFixedMod(pAdicRingBaseGeneric, pAdicFixedModRingGeneric):
     r"""
     An implementation of the `p`-adic integers using fixed modulus.
     """
-    def __init__(self, p, prec, print_mode, names):
+    def __init__(self, p, prec, print_mode, names, category=None):
         """
         Initialization.
 
@@ -550,7 +550,7 @@ def __init__(self, p, prec, print_mode, names):
             sage: K(R(90))
             3*5 + 3*5^2
         """
-        pAdicRingBaseGeneric.__init__(self, p, prec, print_mode, names, pAdicFixedModElement)
+        pAdicRingBaseGeneric.__init__(self, p, prec, print_mode, names, pAdicFixedModElement, category=category)
 
     def _coerce_map_from_(self, R):
         """
@@ -618,7 +618,7 @@ class pAdicFieldCappedRelative(pAdicFieldBaseGeneric, pAdicCappedRelativeFieldGe
         sage: K = Qp(101)  # indirect doctest
     """
 
-    def __init__(self, p, prec, print_mode, names):
+    def __init__(self, p, prec, print_mode, names, category=None):
         """
         Initialization.
 
@@ -660,7 +660,7 @@ def __init__(self, p, prec, print_mode, names):
             ....:                  max_runs=2^5, skip='_test_log')
             sage: R._test_log(max_runs=2, elements=[R.random_element() for i in range(4)])
         """
-        pAdicFieldBaseGeneric.__init__(self, p, prec, print_mode, names, pAdicCappedRelativeElement)
+        pAdicFieldBaseGeneric.__init__(self, p, prec, print_mode, names, pAdicCappedRelativeElement, category=category)
 
     def _coerce_map_from_(self, R):
         """
@@ -749,7 +749,7 @@ class pAdicFieldFloatingPoint(pAdicFieldBaseGeneric, pAdicFloatingPointFieldGene
     An implementation of the `p`-adic rationals with floating point
     precision.
     """
-    def __init__(self, p, prec, print_mode, names):
+    def __init__(self, p, prec, print_mode, names, category=None):
         """
         Initialization.
 
@@ -786,7 +786,7 @@ def __init__(self, p, prec, print_mode, names):
             ....:                  max_runs=2^5, skip='_test_log')
             sage: R._test_log(max_runs=2, elements=[R.random_element() for i in range(4)])
         """
-        pAdicFieldBaseGeneric.__init__(self, p, prec, print_mode, names, pAdicFloatingPointElement)
+        pAdicFieldBaseGeneric.__init__(self, p, prec, print_mode, names, pAdicFloatingPointElement, category=category)
 
     def _coerce_map_from_(self, R):
         """
@@ -881,7 +881,7 @@ class pAdicRingLattice(pAdicLatticeGeneric, pAdicRingBaseGeneric):
         sage: R
         2-adic Ring with lattice-cap precision (label: init)
     """
-    def __init__(self, p, prec, subtype, print_mode, names, label=None):
+    def __init__(self, p, prec, subtype, print_mode, names, label=None, category=None):
         """
         Initialization.
 
@@ -893,11 +893,11 @@ def __init__(self, p, prec, subtype, print_mode, names, label=None):
         # We need to set the subtype first, so that
         # pAdicRingBaseGeneric.__init__ can work
         self._subtype = subtype
-        if isinstance(prec,tuple):
-            pAdicRingBaseGeneric.__init__(self, p, prec[1], print_mode, names, None)
+        if isinstance(prec, tuple):
+            pAdicRingBaseGeneric.__init__(self, p, prec[1], print_mode, names, None, category=category)
         else:
-            pAdicRingBaseGeneric.__init__(self, p, prec, print_mode, names, None)
-        pAdicLatticeGeneric.__init__(self, p, prec, print_mode, names, label)
+            pAdicRingBaseGeneric.__init__(self, p, prec, print_mode, names, None, category=category)
+        pAdicLatticeGeneric.__init__(self, p, prec, print_mode, names, label, category=category)
 
     def _coerce_map_from_(self, R):
         """
@@ -1012,7 +1012,7 @@ class pAdicFieldLattice(pAdicLatticeGeneric, pAdicFieldBaseGeneric):
         sage: R
         2-adic Field with lattice-cap precision (label: init)
     """
-    def __init__(self, p, prec, subtype, print_mode, names, label=None):
+    def __init__(self, p, prec, subtype, print_mode, names, label=None, category=None):
         """
         Initialization.
 
@@ -1024,11 +1024,11 @@ def __init__(self, p, prec, subtype, print_mode, names, label=None):
         # We need to set the subtype first, so that
         # pAdicFieldBaseGeneric.__init__ can work
         self._subtype = subtype
-        if isinstance(prec,tuple):
-            pAdicFieldBaseGeneric.__init__(self, p, prec[1], print_mode, names, None)
+        if isinstance(prec, tuple):
+            pAdicFieldBaseGeneric.__init__(self, p, prec[1], print_mode, names, None, category=category)
         else:
-            pAdicFieldBaseGeneric.__init__(self, p, prec, print_mode, names, None)
-        pAdicLatticeGeneric.__init__(self, p, prec, print_mode, names, label)
+            pAdicFieldBaseGeneric.__init__(self, p, prec, print_mode, names, None, category=category)
+        pAdicLatticeGeneric.__init__(self, p, prec, print_mode, names, label, category=category)
 
     def _coerce_map_from_(self, R):
         """
@@ -1137,7 +1137,7 @@ class pAdicRingRelaxed(pAdicRelaxedGeneric, pAdicRingBaseGeneric):
         sage: type(R)                                                                   # needs sage.libs.flint
         <class 'sage.rings.padics.padic_base_leaves.pAdicRingRelaxed_with_category'>
     """
-    def __init__(self, p, prec, print_mode, names):
+    def __init__(self, p, prec, print_mode, names, category=None):
         """
         Initialization.
 
@@ -1151,7 +1151,7 @@ def __init__(self, p, prec, print_mode, names):
         """
         from sage.rings.padics import padic_relaxed_element
         self._default_prec, self._halting_prec, self._secure = prec
-        pAdicRingBaseGeneric.__init__(self, p, self._default_prec, print_mode, names, padic_relaxed_element.pAdicRelaxedElement)
+        pAdicRingBaseGeneric.__init__(self, p, self._default_prec, print_mode, names, padic_relaxed_element.pAdicRelaxedElement, category=category)
         self._element_class_module = padic_relaxed_element
         self._element_class_prefix = "pAdicRelaxedElement_"
 
@@ -1176,7 +1176,7 @@ class pAdicFieldRelaxed(pAdicRelaxedGeneric, pAdicFieldBaseGeneric):
         sage: type(R)                                                                   # needs sage.libs.flint
         <class 'sage.rings.padics.padic_base_leaves.pAdicFieldRelaxed_with_category'>
     """
-    def __init__(self, p, prec, print_mode, names):
+    def __init__(self, p, prec, print_mode, names, category=None):
         """
         Initialization.
 
@@ -1190,6 +1190,6 @@ def __init__(self, p, prec, print_mode, names):
         """
         from sage.rings.padics import padic_relaxed_element
         self._default_prec, self._halting_prec, self._secure = prec
-        pAdicFieldBaseGeneric.__init__(self, p, self._default_prec, print_mode, names, padic_relaxed_element.pAdicRelaxedElement)
+        pAdicFieldBaseGeneric.__init__(self, p, self._default_prec, print_mode, names, padic_relaxed_element.pAdicRelaxedElement, category=category)
         self._element_class_module = padic_relaxed_element
         self._element_class_prefix = "pAdicRelaxedElement_"