gh-35283: Modularization fixes for imports of number fields
    
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### 📚 Description

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Using the ABC `sage.rings.number_field.number_field_base.NumberField`

Deprecating `is_NumberField`.

Part of:
- #29705

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- [ ] I have updated the documentation accordingly.

### ⌛ Dependencies
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- Depends on #35277
    
URL: #35283
Reported by: Matthias Köppe
Reviewer(s): Marc Mezzarobba

Author: Release Manager

src/sage/algebras/quatalg/quaternion_algebra.py

@@ -51,7 +51,7 @@
 from sage.rings.ideal import Ideal_fractional
 from sage.rings.rational_field import is_RationalField, QQ
 from sage.rings.infinity import infinity
-from sage.rings.number_field.number_field import is_NumberField
+from sage.rings.number_field.number_field_base import NumberField
 from sage.rings.power_series_ring import PowerSeriesRing
 from sage.structure.category_object import normalize_names
 from sage.structure.parent import Parent
@@ -662,7 +662,7 @@ def __init__(self, base_ring, a, b, names='i,j,k'):
         self._b = b
         if is_RationalField(base_ring) and a.denominator() == 1 == b.denominator():
             self.Element = QuaternionAlgebraElement_rational_field
-        elif (is_NumberField(base_ring) and base_ring.degree() > 2 and base_ring.is_absolute() and
+        elif (isinstance(base_ring, NumberField) and base_ring.degree() > 2 and base_ring.is_absolute() and
               a.denominator() == 1 == b.denominator() and base_ring.defining_polynomial().is_monic()):
             # This QuaternionAlgebraElement_number_field class is not
             # designed to work with elements of a quadratic field.  To

src/sage/categories/number_fields.py

@@ -89,8 +89,8 @@ def __contains__(self, x):
             sage: ZZ in NumberFields()
             False
         """
-        import sage.rings.number_field.number_field_base
-        return sage.rings.number_field.number_field_base.is_NumberField(x)
+        from sage.rings.number_field.number_field_base import NumberField
+        return isinstance(x, NumberField)
 
     def _call_(self, x):
         r"""

src/sage/matrix/matrix2.pyx

@@ -89,7 +89,7 @@ from sage.misc.verbose import verbose, get_verbose
 from sage.categories.fields import Fields
 from sage.categories.integral_domains import IntegralDomains
 from sage.rings.ring import is_Ring
-from sage.rings.number_field.number_field_base import is_NumberField
+from sage.rings.number_field.number_field_base import NumberField
 from sage.rings.integer_ring import ZZ, is_IntegerRing
 from sage.rings.integer import Integer
 from sage.rings.rational_field import QQ, is_RationalField
@@ -4457,7 +4457,7 @@ cdef class Matrix(Matrix1):
             raise ValueError("'padic' matrix kernel algorithm only available over the rationals and the integers, not over %s" % R)
         elif algorithm == 'flint' and not (is_IntegerRing(R) or is_RationalField(R)):
             raise ValueError("'flint' matrix kernel algorithm only available over the rationals and the integers, not over %s" % R)
-        elif algorithm == 'pari' and not (is_IntegerRing(R) or (is_NumberField(R) and not is_RationalField(R))):
+        elif algorithm == 'pari' and not (is_IntegerRing(R) or (isinstance(R, NumberField) and not is_RationalField(R))):
             raise ValueError("'pari' matrix kernel algorithm only available over non-trivial number fields and the integers, not over %s" % R)
         elif algorithm == 'generic' and R not in _Fields:
             raise ValueError("'generic' matrix kernel algorithm only available over a field, not over %s" % R)
@@ -4514,7 +4514,7 @@ cdef class Matrix(Matrix1):
             except AttributeError:
                 pass
 
-        if M is None and is_NumberField(R):
+        if M is None and isinstance(R, NumberField):
             format, M = self._right_kernel_matrix_over_number_field()
 
         if M is None and R in _Fields:

src/sage/modular/dirichlet.py

@@ -60,7 +60,6 @@
 import sage.misc.prandom as random
 import sage.modules.free_module_element as free_module_element
 import sage.rings.abc
-import sage.rings.number_field.number_field as number_field
 
 from sage.arith.functions import lcm
 from sage.arith.misc import bernoulli, kronecker, factor, gcd, fundamental_discriminant, euler_phi, valuation
@@ -77,7 +76,7 @@
 from sage.rings.finite_rings.integer_mod_ring import IntegerModRing
 from sage.rings.integer import Integer
 from sage.rings.integer_ring import ZZ
-from sage.rings.number_field.number_field import CyclotomicField
+from sage.rings.number_field.number_field import CyclotomicField, NumberField, NumberField_generic
 from sage.rings.power_series_ring import PowerSeriesRing
 from sage.rings.rational_field import RationalField, QQ, is_RationalField
 from sage.rings.ring import is_Ring
@@ -1039,7 +1038,7 @@ def fixed_field(self):
             sage: psi.fixed_field()
             Number Field in a with defining polynomial x^5 + x^4 - 12*x^3 - 21*x^2 + x + 5
         """
-        return number_field.NumberField(self.fixed_field_polynomial(), 'a')
+        return NumberField(self.fixed_field_polynomial(), 'a')
 
     @cached_method
     def decomposition(self):
@@ -1903,7 +1902,7 @@ def minimize_base_ring(self):
             K = IntegerModRing(p)
         elif self.order() <= 2:
             K = QQ
-        elif (isinstance(R, number_field.NumberField_generic)
+        elif (isinstance(R, NumberField_generic)
               and euler_phi(self.order()) < R.absolute_degree()):
             K = CyclotomicField(self.order())
         else:

src/sage/modular/modsym/space.py

@@ -35,7 +35,7 @@
 from sage.rings.integer_ring import ZZ
 from sage.rings.integer import Integer
 from sage.rings.infinity import infinity
-from sage.rings.number_field.number_field_base import is_NumberField
+from sage.rings.number_field.number_field_base import NumberField
 from sage.rings.power_series_ring import PowerSeriesRing
 from sage.rings.rational_field import QQ
 from sage.structure.all import Sequence, SageObject
@@ -1046,7 +1046,7 @@ def _q_expansion_module_rational(self, prec):
         if not self.is_cuspidal():
             raise ValueError("self must be cuspidal")
         K = self.base_ring()
-        if not is_NumberField(K):
+        if not isinstance(K, NumberField):
             raise TypeError("self must be over QQ or a number field.")
         n = K.degree()
         if n == 1:

src/sage/rings/morphism.pyx

@@ -3162,7 +3162,7 @@ def _tensor_product_ring(B, A):
         ValueError: term ordering must be global
     """
     from .finite_rings.finite_field_base import FiniteField
-    from .number_field.number_field_base import is_NumberField
+    from .number_field.number_field_base import NumberField
     from .polynomial.multi_polynomial_ring import is_MPolynomialRing
     from .polynomial.polynomial_quotient_ring import is_PolynomialQuotientRing
     from .polynomial.polynomial_ring import is_PolynomialRing
@@ -3179,7 +3179,7 @@ def _tensor_product_ring(B, A):
     def term_order(A):
         # univariate rings do not have a term order
         if (is_PolynomialRing(A) or is_PolynomialQuotientRing(A)
-            or ((is_NumberField(A) or isinstance(A, FiniteField))
+            or (isinstance(A, (NumberField, FiniteField))
                 and not A.is_prime_field())):
             return TermOrder('lex', 1)
         try:
@@ -3202,7 +3202,7 @@ def _tensor_product_ring(B, A):
         elif is_QuotientRing(A):
             to_R = A.ambient().hom(R_gens_A, R, check=False)
             return list(to_R(A.defining_ideal()).gens())
-        elif ((is_NumberField(A) or isinstance(A, FiniteField))
+        elif (isinstance(A, (NumberField, FiniteField))
               and not A.is_prime_field()):
             to_R = A.polynomial_ring().hom(R_gens_A, R, check=False)
             return [to_R(A.polynomial())]

src/sage/rings/number_field/number_field.py

@@ -918,10 +918,10 @@ def QuadraticField(D, name='a', check=True, embedding=True, latex_name='sqrt', *
 
     ::
 
-        sage: from sage.rings.number_field.number_field import is_NumberField
+        sage: from sage.rings.number_field.number_field_base import NumberField
         sage: type(K)
         <class 'sage.rings.number_field.number_field.NumberField_quadratic_with_category'>
-        sage: is_NumberField(K)
+        sage: isinstance(K, NumberField)
         True
 
     Quadratic number fields are cached::
@@ -8308,7 +8308,7 @@ def _coerce_map_from_(self, R):
         if is_NumberFieldOrder(R) and self.has_coerce_map_from(R.number_field()):
             return self._generic_coerce_map(R)
         # R is not QQ by the above tests
-        if is_NumberField(R) and R.coerce_embedding() is not None:
+        if isinstance(R, number_field_base.NumberField) and R.coerce_embedding() is not None:
             if self.coerce_embedding() is not None:
                 try:
                     return number_field_morphisms.EmbeddedNumberFieldMorphism(R, self)

src/sage/rings/number_field/number_field_base.pyx

@@ -11,29 +11,41 @@ TESTS::
 
 def is_NumberField(x):
     """
-    Return True if x is of number field type.
+    Return True if ``x`` is of number field type.
+
+    This function is deprecated.
 
     EXAMPLES::
 
         sage: from sage.rings.number_field.number_field_base import is_NumberField
-        sage: is_NumberField(NumberField(x^2+1,'a'))
+        sage: is_NumberField(NumberField(x^2 + 1, 'a'))
+        doctest:...: DeprecationWarning: the function is_NumberField is deprecated; use
+        isinstance(x, sage.rings.number_field.number_field_base.NumberField) instead
+        See https://github.com/sagemath/sage/issues/35283 for details.
         True
-        sage: is_NumberField(QuadraticField(-97,'theta'))
+        sage: is_NumberField(QuadraticField(-97, 'theta'))
         True
         sage: is_NumberField(CyclotomicField(97))
         True
 
-    Note that the rational numbers QQ are a number field.::
+    Note that the rational numbers ``QQ`` are a number field.::
 
         sage: is_NumberField(QQ)
         True
         sage: is_NumberField(ZZ)
         False
     """
+    from sage.misc.superseded import deprecation
+    deprecation(35283,
+                "the function is_NumberField is deprecated; use "
+                "isinstance(x, sage.rings.number_field.number_field_base.NumberField) instead")
+
     return isinstance(x, NumberField)
 
+
 from sage.rings.ring cimport Field
 
+
 cdef class NumberField(Field):
     r"""
     Base class for all number fields.

src/sage/rings/number_field/number_field_element.pyx

@@ -1672,8 +1672,8 @@ cdef class NumberFieldElement(NumberFieldElement_base):
             return self.is_norm(L, element=True, proof=proof)[0]
 
         K = self.parent()
-        from sage.rings.number_field.number_field_base import is_NumberField
-        if not is_NumberField(L):
+        from sage.rings.number_field.number_field_base import NumberField
+        if not isinstance(L, NumberField):
             raise ValueError("L (=%s) must be a NumberField in is_norm" % L)
 
         from sage.rings.number_field.number_field import is_AbsoluteNumberField
@@ -3812,8 +3812,8 @@ cdef class NumberFieldElement(NumberFieldElement_base):
         if base is self.parent():
             return MatrixSpace(base,1)([self])
         if base is not None and base is not self.base_ring():
-            from sage.rings.number_field.number_field_base import is_NumberField
-            if is_NumberField(base):
+            from sage.rings.number_field.number_field_base import NumberField
+            if isinstance(base, NumberField):
                 return self._matrix_over_base(base)
             else:
                 return self._matrix_over_base_morphism(base)

src/sage/rings/number_field/number_field_rel.py

@@ -93,7 +93,7 @@
 from .number_field import (NumberField, NumberField_generic,
     put_natural_embedding_first, proof_flag,
     is_NumberFieldHomsetCodomain)
-from sage.rings.number_field.number_field_base import is_NumberField
+from sage.rings.number_field.number_field_base import NumberField as NumberField_base
 from sage.rings.number_field.order import (RelativeOrder, is_NumberFieldOrder,
                                            relative_order_from_ring_generators)
 from sage.rings.number_field.morphism import RelativeNumberFieldHomomorphism_from_abs
@@ -277,7 +277,7 @@ def __init__(self, base, polynomial, name,
             raise NotImplementedError("Embeddings not implemented for relative number fields")
         if names is not None:
             name = names
-        if not is_NumberField(base):
+        if not isinstance(base, NumberField_base):
             raise TypeError("base (=%s) must be a number field"%base)
         if not isinstance(polynomial, polynomial_element.Polynomial):
             try: