gh-37069: Use Parent in number fields and QQ
    
Towards the removal of `ParentWithGens` : use parent in number fields
and in QQ

### 📝 Checklist

- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
    
URL: #37069
Reported by: Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/rings/number_field/number_field.py

@@ -77,8 +77,7 @@
 # ****************************************************************************
 from __future__ import annotations
 from sage.misc.cachefunc import cached_method
-from sage.misc.superseded import (deprecation,
-                                  deprecated_function_alias)
+from sage.misc.superseded import deprecation
 
 
 import sage.libs.ntl.all as ntl
@@ -109,6 +108,7 @@
 from .class_group import SClassGroup
 
 from sage.structure.element import is_Element
+from sage.structure.parent import Parent
 from sage.structure.sequence import Sequence
 from sage.structure.factorization import Factorization
 from sage.structure.category_object import normalize_names
@@ -209,7 +209,6 @@ def proof_flag(t):
 import sage.groups.abelian_gps.abelian_group
 import sage.rings.complex_interval_field
 
-from sage.structure.parent_gens import ParentWithGens
 from sage.structure.factory import UniqueFactory
 from . import number_field_element
 from . import number_field_element_quadratic
@@ -1379,7 +1378,7 @@ def __init__(self, polynomial, name, latex_name,
         else:
             assert category.is_subcategory(default_category), "%s is not a subcategory of %s" % (category, default_category)
 
-        ParentWithGens.__init__(self, QQ, name, category=category)
+        Parent.__init__(self, base=QQ, names=name, category=category)
         if not isinstance(polynomial, polynomial_element.Polynomial):
             raise TypeError("polynomial (=%s) must be a polynomial" % repr(polynomial))
 

src/sage/rings/rational_field.py

@@ -62,7 +62,7 @@
 import sage.rings.number_field.number_field_base as number_field_base
 from sage.misc.fast_methods import Singleton
 from sage.misc.superseded import deprecated_function_alias
-from sage.structure.parent_gens import ParentWithGens
+from sage.structure.parent import Parent
 from sage.structure.sequence import Sequence
 
 
@@ -235,8 +235,9 @@ def __init__(self):
         """
         from sage.categories.basic import QuotientFields
         from sage.categories.number_fields import NumberFields
-        ParentWithGens.__init__(self, self, category=[QuotientFields().Metric(),
-                                                      NumberFields()])
+        Parent.__init__(self, base=self,
+                        category=[QuotientFields().Metric(),
+                                  NumberFields()])
         self._assign_names(('x',), normalize=False)  # ?????
         self._populate_coercion_lists_(init_no_parent=True)
 
@@ -862,7 +863,7 @@ def hilbert_symbol_negative_at_S(self, S, b, check=True):
             if p != infty:
                 if check and not is_prime(p):
                     raise ValueError("all entries in list must be prime"
-                                    " or -1 for infinite place")
+                                     " or -1 for infinite place")
                 R = Qp(p)
                 if R(b).is_square():
                     raise ValueError("second argument must be a nonsquare with"
@@ -1055,7 +1056,7 @@ def power_basis(self):
             sage: QQ.power_basis()
             [1]
         """
-        return [ self.gen() ]
+        return [self.gen()]
 
     def extension(self, poly, names, **kwds):
         r"""
@@ -1133,7 +1134,7 @@ def _an_element_(self):
             sage: QQ.an_element() # indirect doctest
             1/2
         """
-        return Rational((1,2))
+        return Rational((1, 2))
 
     def some_elements(self):
         r"""
@@ -1247,7 +1248,7 @@ def random_element(self, num_bound=None, den_bound=None, *args, **kwds):
             den = ZZ.random_element(1, den_bound+1, *args, **kwds)
             while den == 0:
                 den = ZZ.random_element(1, den_bound+1, *args, **kwds)
-            return self((num,den))
+            return self((num, den))
 
     def zeta(self, n=2):
         """
@@ -1381,7 +1382,7 @@ def selmer_group_iterator(self, S, m, proof=True):
 
         from sage.misc.misc_c import prod
         for ev in product(*[range(o) for o in ords]):
-            yield prod((p**e for p,e in zip(KSgens, ev)), one)
+            yield prod((p**e for p, e in zip(KSgens, ev)), one)
 
     def selmer_space(self, S, p, proof=None):
         r"""
@@ -1520,7 +1521,7 @@ def quadratic_defect(self, a, p, check=True):
                 return v + 1
 
     #################################
-    ## Coercions to interfaces
+    #  Coercions to interfaces
     #################################
     def _gap_init_(self):
         r"""
@@ -1658,12 +1659,12 @@ def _factor_univariate_polynomial(self, f):
             (10) * (x^5 - 1/10)
             sage: QQ._factor_univariate_polynomial(10*x^5 - 10)
             (10) * (x - 1) * (x^4 + x^3 + x^2 + x + 1)
-
         """
-        G = list(f._pari_with_name().factor())
+        G = f._pari_with_name().factor()
 
         # normalize the leading coefficients
-        F = [(f.parent()(g).monic(), int(e)) for (g,e) in zip(*G)]
+        P = f.parent()
+        F = [(P(g).monic(), int(e)) for g, e in zip(*G)]
 
         from sage.structure.factorization import Factorization
         return Factorization(F, f.leading_coefficient())