gh-36865: `sage.groups`, `sage.rings.number_field`: Modularization fixes, `# needs`
    
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- Cherry-picked from #35095

Standard modularization work:
- a new dynamically detected feature `sage.libs.braiding`
- using file-level and block-level `# needs` to simplify decorations
- adding some `# needs`, in particular for `sage.libs.gap`
- splitting GAP-dependent classes out from `sage.groups.galois_group`
- using more `lazy_import`s
- some doctest cosmetics

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example "Fixes #12345". -->
Part of:
- #29705
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- [ ] I have linked a relevant issue or discussion.
- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation accordingly.

### ⌛ Dependencies

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URL: #36865
Reported by: Matthias Köppe
Reviewer(s): David Coudert

Author: Release Manager

src/sage/crypto/lattice.py

@@ -211,7 +211,7 @@ def gen_lattice(type='modular', n=4, m=8, q=11, seed=None,
         [-4 -3 2 -5 0 0 0 0 0 1]
         ]
 
-        sage: sage.crypto.gen_lattice(m=10, q=11, seed=42, lattice=True)
+        sage: sage.crypto.gen_lattice(m=10, q=11, seed=42, lattice=True)                # needs fpylll
         Free module of degree 10 and rank 10 over Integer Ring
         User basis matrix:
         [ 0  0  1  1  0 -1 -1 -1  1  0]

src/sage/features/sagemath.py

@@ -311,6 +311,29 @@ def __init__(self):
                              spkg='sagemath_groups', type='standard')
 
 
+class sage__libs__braiding(PythonModule):
+    r"""
+    A :class:`~sage.features.Feature` describing the presence of :mod:`sage.libs.braiding`.
+
+    EXAMPLES::
+
+        sage: from sage.features.sagemath import sage__libs__braiding
+        sage: sage__libs__braiding().is_present()                                            # needs sage.libs.braiding
+        FeatureTestResult('sage.libs.braiding', True)
+    """
+
+    def __init__(self):
+        r"""
+        TESTS::
+
+            sage: from sage.features.sagemath import sage__libs__braiding
+            sage: isinstance(sage__libs__braiding(), sage__libs__braiding)
+            True
+        """
+        PythonModule.__init__(self, 'sage.libs.braiding',
+                              spkg='sagemath_libbraiding', type='standard')
+
+
 class sage__libs__ecl(PythonModule):
     r"""
     A :class:`~sage.features.Feature` describing the presence of :mod:`sage.libs.ecl`.
@@ -330,7 +353,8 @@ def __init__(self):
             sage: isinstance(sage__libs__ecl(), sage__libs__ecl)
             True
         """
-        PythonModule.__init__(self, 'sage.libs.ecl')
+        PythonModule.__init__(self, 'sage.libs.ecl',
+                              spkg='sagemath_symbolics', type='standard')
 
 
 class sage__libs__flint(JoinFeature):
@@ -1076,6 +1100,7 @@ def all_features():
             sage__geometry__polyhedron(),
             sage__graphs(),
             sage__groups(),
+            sage__libs__braiding(),
             sage__libs__ecl(),
             sage__libs__flint(),
             sage__libs__gap(),

src/sage/groups/abelian_gps/dual_abelian_group.py

@@ -1,3 +1,4 @@
+# sage.doctest: needs sage.rings.number_field
 r"""
 Dual groups of Finite Multiplicative Abelian Groups
 
@@ -25,7 +26,6 @@
     sage: F = AbelianGroup(5, [2,5,7,8,9], names='abcde')
     sage: (a, b, c, d, e) = F.gens()
 
-    sage: # needs sage.rings.number_field
     sage: Fd = F.dual_group(names='ABCDE')
     sage: Fd.base_ring()
     Cyclotomic Field of order 2520 and degree 576
@@ -82,7 +82,6 @@ def is_DualAbelianGroup(x):
 
     EXAMPLES::
 
-        sage: # needs sage.rings.number_field
         sage: from sage.groups.abelian_gps.dual_abelian_group import is_DualAbelianGroup
         sage: F = AbelianGroup(5,[3,5,7,8,9], names=list("abcde"))
         sage: Fd = F.dual_group()
@@ -105,7 +104,7 @@ class DualAbelianGroup_class(UniqueRepresentation, AbelianGroupBase):
     EXAMPLES::
 
         sage: F = AbelianGroup(5,[3,5,7,8,9], names="abcde")
-        sage: F.dual_group()                                                            # needs sage.rings.number_field
+        sage: F.dual_group()
         Dual of Abelian Group isomorphic to Z/3Z x Z/5Z x Z/7Z x Z/8Z x Z/9Z
         over Cyclotomic Field of order 2520 and degree 576
 
@@ -123,7 +122,7 @@ def __init__(self, G, names, base_ring):
         EXAMPLES::
 
             sage: F = AbelianGroup(5,[3,5,7,8,9], names="abcde")
-            sage: F.dual_group()                                                        # needs sage.rings.number_field
+            sage: F.dual_group()
             Dual of Abelian Group isomorphic to Z/3Z x Z/5Z x Z/7Z x Z/8Z x Z/9Z
             over Cyclotomic Field of order 2520 and degree 576
        """
@@ -180,9 +179,9 @@ def _repr_(self):
         EXAMPLES::
 
             sage: F = AbelianGroup(5, [2,5,7,8,9], names='abcde')
-            sage: Fd = F.dual_group(names='ABCDE',                                      # needs sage.rings.number_field
+            sage: Fd = F.dual_group(names='ABCDE',
             ....:                   base_ring=CyclotomicField(2*5*7*8*9))
-            sage: Fd   # indirect doctest                                               # needs sage.rings.number_field
+            sage: Fd   # indirect doctest
             Dual of Abelian Group isomorphic to Z/2Z x Z/5Z x Z/7Z x Z/8Z x Z/9Z
             over Cyclotomic Field of order 5040 and degree 1152
             sage: Fd = F.dual_group(names='ABCDE', base_ring=CC)                        # needs sage.rings.real_mpfr
@@ -209,8 +208,8 @@ def _latex_(self):
         EXAMPLES::
 
             sage: F = AbelianGroup(3, [2]*3)
-            sage: Fd = F.dual_group()                                                   # needs sage.rings.number_field
-            sage: Fd._latex_()                                                          # needs sage.rings.number_field
+            sage: Fd = F.dual_group()
+            sage: Fd._latex_()
             '$\\mathrm{DualAbelianGroup}( AbelianGroup ( 3, (2, 2, 2) ) )$'
         """
         return r"$\mathrm{DualAbelianGroup}( AbelianGroup ( %s, %s ) )$" % (self.ngens(), self.gens_orders())
@@ -251,7 +250,6 @@ def gen(self, i=0):
 
         EXAMPLES::
 
-            sage: # needs sage.rings.number_field
             sage: F = AbelianGroup(3, [1,2,3], names='a')
             sage: Fd = F.dual_group(names="A")
             sage: Fd.0
@@ -279,8 +277,8 @@ def gens(self):
 
         EXAMPLES::
 
-            sage: F = AbelianGroup([7,11]).dual_group()                                 # needs sage.rings.number_field
-            sage: F.gens()                                                              # needs sage.rings.number_field
+            sage: F = AbelianGroup([7,11]).dual_group()
+            sage: F.gens()
             (X0, X1)
         """
         n = self.group().ngens()
@@ -293,8 +291,8 @@ def ngens(self):
         EXAMPLES::
 
             sage: F = AbelianGroup([7]*100)
-            sage: Fd = F.dual_group()                                                   # needs sage.rings.number_field
-            sage: Fd.ngens()                                                            # needs sage.rings.number_field
+            sage: Fd = F.dual_group()
+            sage: Fd.ngens()
             100
         """
         return self.group().ngens()
@@ -310,8 +308,8 @@ def gens_orders(self):
         EXAMPLES::
 
             sage: F = AbelianGroup([5]*1000)
-            sage: Fd = F.dual_group()                                                   # needs sage.rings.number_field
-            sage: invs = Fd.gens_orders(); len(invs)                                    # needs sage.rings.number_field
+            sage: Fd = F.dual_group()
+            sage: invs = Fd.gens_orders(); len(invs)
             1000
         """
         return self.group().gens_orders()
@@ -325,8 +323,8 @@ def invariants(self):
         EXAMPLES::
 
             sage: F = AbelianGroup([5]*1000)
-            sage: Fd = F.dual_group()                                                   # needs sage.rings.number_field
-            sage: invs = Fd.gens_orders(); len(invs)                                    # needs sage.rings.number_field
+            sage: Fd = F.dual_group()
+            sage: invs = Fd.gens_orders(); len(invs)
             1000
         """
         # TODO: deprecate
@@ -340,9 +338,9 @@ def __contains__(self, X):
 
             sage: F = AbelianGroup(5,[2, 3, 5, 7, 8], names="abcde")
             sage: a,b,c,d,e = F.gens()
-            sage: Fd = F.dual_group(names="ABCDE")                                      # needs sage.rings.number_field
-            sage: A,B,C,D,E = Fd.gens()                                                 # needs sage.rings.number_field
-            sage: A*B^2*D^7 in Fd                                                       # needs sage.rings.number_field
+            sage: Fd = F.dual_group(names="ABCDE")
+            sage: A,B,C,D,E = Fd.gens()
+            sage: A*B^2*D^7 in Fd
             True
         """
         return X.parent() == self and is_DualAbelianGroupElement(X)
@@ -354,8 +352,8 @@ def order(self):
         EXAMPLES::
 
             sage: G = AbelianGroup([2,3,9])
-            sage: Gd = G.dual_group()                                                   # needs sage.rings.number_field
-            sage: Gd.order()                                                            # needs sage.rings.number_field
+            sage: Gd = G.dual_group()
+            sage: Gd.order()
             54
         """
         G = self.group()
@@ -368,10 +366,10 @@ def is_commutative(self):
         EXAMPLES::
 
             sage: G = AbelianGroup([2,3,9])
-            sage: Gd = G.dual_group()                                                   # needs sage.rings.number_field
-            sage: Gd.is_commutative()                                                   # needs sage.rings.number_field
+            sage: Gd = G.dual_group()
+            sage: Gd.is_commutative()
             True
-            sage: Gd.is_abelian()                                                       # needs sage.rings.number_field
+            sage: Gd.is_abelian()
             True
         """
         return True
@@ -384,8 +382,8 @@ def list(self):
         EXAMPLES::
 
             sage: G = AbelianGroup([2,3], names="ab")
-            sage: Gd = G.dual_group(names="AB")                                         # needs sage.rings.number_field
-            sage: Gd.list()                                                             # needs sage.rings.number_field
+            sage: Gd = G.dual_group(names="AB")
+            sage: Gd.list()
             (1, B, B^2, A, A*B, A*B^2)
         """
         if not self.is_finite():
@@ -400,8 +398,8 @@ def __iter__(self):
         EXAMPLES::
 
             sage: G = AbelianGroup([2,3], names="ab")
-            sage: Gd = G.dual_group(names="AB")                                         # needs sage.rings.number_field
-            sage: [X for X in Gd]                                                       # needs sage.rings.number_field
+            sage: Gd = G.dual_group(names="AB")
+            sage: [X for X in Gd]
             [1, B, B^2, A, A*B, A*B^2]
 
             sage: # needs sage.rings.real_mpfr

src/sage/groups/abelian_gps/dual_abelian_group_element.py

@@ -1,3 +1,4 @@
+# sage.doctest: needs sage.rings.number_field
 """
 Elements (characters) of the dual group of a finite Abelian group
 
@@ -8,13 +9,12 @@
     sage: F
     Multiplicative Abelian group isomorphic to C2 x C3 x C5 x C7 x C8
 
-    sage: Fd = F.dual_group(names="ABCDE"); Fd                                          # needs sage.rings.number_field
+    sage: Fd = F.dual_group(names="ABCDE"); Fd
     Dual of Abelian Group isomorphic to Z/2Z x Z/3Z x Z/5Z x Z/7Z x Z/8Z
     over Cyclotomic Field of order 840 and degree 192
 
 The elements of the dual group can be evaluated on elements of the original group::
 
-    sage: # needs sage.rings.number_field
     sage: a,b,c,d,e = F.gens()
     sage: A,B,C,D,E = Fd.gens()
     sage: A*B^2*D^7
@@ -71,10 +71,10 @@ def is_DualAbelianGroupElement(x) -> bool:
     EXAMPLES::
 
         sage: from sage.groups.abelian_gps.dual_abelian_group import is_DualAbelianGroupElement
-        sage: F = AbelianGroup(5, [5,5,7,8,9], names=list("abcde")).dual_group()        # needs sage.rings.number_field
-        sage: is_DualAbelianGroupElement(F)                                             # needs sage.rings.number_field
+        sage: F = AbelianGroup(5, [5,5,7,8,9], names=list("abcde")).dual_group()
+        sage: is_DualAbelianGroupElement(F)
         False
-        sage: is_DualAbelianGroupElement(F.an_element())                                # needs sage.rings.number_field
+        sage: is_DualAbelianGroupElement(F.an_element())
         True
     """
     return isinstance(x, DualAbelianGroupElement)
@@ -96,7 +96,6 @@ def __call__(self, g):
 
         EXAMPLES::
 
-            sage: # needs sage.rings.number_field
             sage: F = AbelianGroup(5, [2,3,5,7,8], names="abcde")
             sage: a,b,c,d,e = F.gens()
             sage: Fd = F.dual_group(names="ABCDE")
@@ -147,7 +146,6 @@ def word_problem(self, words):
 
         EXAMPLES::
 
-            sage: # needs sage.rings.number_field
             sage: G = AbelianGroup(5,[3, 5, 5, 7, 8], names="abcde")
             sage: Gd = G.dual_group(names="abcde")
             sage: a,b,c,d,e = Gd.gens()
@@ -156,7 +154,7 @@ def word_problem(self, words):
             sage: w = a^7*b^3*c^5*d^4*e^4
             sage: x = a^3*b^2*c^2*d^3*e^5
             sage: y = a^2*b^4*c^2*d^4*e^5
-            sage: e.word_problem([u,v,w,x,y])
+            sage: e.word_problem([u,v,w,x,y])                                           # needs sage.libs.gap
             [[b^2*c^2*d^3*e^5, 245]]
         """
         from sage.libs.gap.libgap import libgap

src/sage/groups/affine_gps/affine_group.py

@@ -204,7 +204,10 @@ def __init__(self, degree, ring):
 
             sage: G = AffineGroup(2, GF(5)); G
             Affine Group of degree 2 over Finite Field of size 5
+
+            sage: # needs sage.libs.gap (for gens)
             sage: TestSuite(G).run()
+
             sage: G.category()
             Category of finite groups
 
@@ -289,8 +292,10 @@ def cardinality(self):
 
         EXAMPLES::
 
+            sage: # needs sage.libs.gap
             sage: AffineGroup(6, GF(5)).cardinality()
             172882428468750000000000000000
+
             sage: AffineGroup(6, ZZ).cardinality()
             +Infinity
         """
@@ -464,6 +469,7 @@ def random_element(self):
 
         EXAMPLES::
 
+            sage: # needs sage.libs.gap
             sage: G = AffineGroup(4, GF(3))
             sage: G.random_element()  # random
                   [2 0 1 2]     [1]
@@ -498,6 +504,7 @@ def some_elements(self):
 
         EXAMPLES::
 
+            sage: # needs sage.libs.gap
             sage: G = AffineGroup(4,5)
             sage: G.some_elements()
             [      [2 0 0 0]     [1]

src/sage/groups/affine_gps/euclidean_group.py

@@ -146,6 +146,8 @@ class EuclideanGroup(AffineGroup):
         True
         sage: G = EuclideanGroup(2, GF(5)); G
         Euclidean Group of degree 2 over Finite Field of size 5
+
+        sage: # needs sage.libs.gap (for gens)
         sage: TestSuite(G).run()
 
     REFERENCES:

src/sage/groups/affine_gps/group_element.py

@@ -78,11 +78,14 @@ class AffineGroupElement(MultiplicativeGroupElement):
     EXAMPLES::
 
         sage: G = AffineGroup(2, GF(3))
+
+        sage: # needs sage.libs.gap
         sage: g = G.random_element()
         sage: type(g)
         <class 'sage.groups.affine_gps.affine_group.AffineGroup_with_category.element_class'>
         sage: G(g.matrix()) == g
         True
+
         sage: G(2)
               [2 0]     [0]
         x |-> [0 2] x + [0]
@@ -107,6 +110,7 @@ def __init__(self, parent, A, b=0, convert=True, check=True):
 
         TESTS::
 
+            sage: # needs sage.libs.gap
             sage: G = AffineGroup(4, GF(5))
             sage: g = G.random_element()
             sage: TestSuite(g).run()
@@ -200,6 +204,7 @@ def matrix(self):
         Composition of affine group elements equals multiplication of
         the matrices::
 
+            sage: # needs sage.libs.gap
             sage: g1 = G.random_element()
             sage: g2 = G.random_element()
             sage: g1.matrix() * g2.matrix() == (g1*g2).matrix()

src/sage/groups/braid.py

@@ -72,15 +72,14 @@
 from sage.combinat.permutation import Permutation
 from sage.combinat.permutation import Permutations
 from sage.combinat.subset import Subsets
-from sage.features import PythonModule
+from sage.features.sagemath import sage__libs__braiding
 from sage.groups.artin import FiniteTypeArtinGroup, FiniteTypeArtinGroupElement
 from sage.groups.finitely_presented import FinitelyPresentedGroup
 from sage.groups.finitely_presented import GroupMorphismWithGensImages
 from sage.groups.free_group import FreeGroup, is_FreeGroup
 from sage.functions.generalized import sign
 from sage.groups.perm_gps.permgroup_named import SymmetricGroup
 from sage.groups.perm_gps.permgroup_named import SymmetricGroupElement
-from sage.knots.knot import Knot
 from sage.libs.gap.libgap import libgap
 from sage.matrix.constructor import identity_matrix, matrix
 from sage.misc.lazy_attribute import lazy_attribute
@@ -98,7 +97,8 @@
             ['leftnormalform', 'rightnormalform', 'centralizer', 'supersummitset', 'greatestcommondivisor',
              'leastcommonmultiple', 'conjugatingbraid', 'ultrasummitset',
              'thurston_type', 'rigidity', 'sliding_circuits'],
-            feature=PythonModule('sage.libs.braiding', spkg='libbraiding', type='standard'))
+            feature=sage__libs__braiding())
+lazy_import('sage.knots.knot', 'Knot')
 
 
 class Braid(FiniteTypeArtinGroupElement):