sagemathgh-36044: `sage.rings.number_field`: Modularization fixes, doctest cosmetics, `# needs`
    
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- Avoiding going through symbolic square roots in `.S_unit_solver`
- `# needs sage.symbolic`, `# needs sage.libs.*`, `# needs
sage.rings.finite_rings` etc.
- Reformatting doctests to fit on the screen
- Codestyle fixes in doctests
- Block tags for `# optional - magma`
- Handle import errors in `NumberField._pushout_`

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This is
- Part of: sagemath#29705
- Cherry-picked from: sagemath#35095
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- [ ] I have created tests covering the changes.
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### ⌛ Dependencies

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URL: sagemath#36044
Reported by: Matthias Köppe
Reviewer(s): David Coudert, Kwankyu Lee, Matthias Köppe

Author: Release Manager

src/sage/rings/number_field/S_unit_solver.py

@@ -982,11 +982,12 @@ def minimal_vector(A, y, prec=106):
     ALLLinv = ALLL.inverse()
     ybrace = [ abs(R(a-a.round())) for a in y * ALLLinv if (a-a.round()) != 0]
 
+    v = ALLL.rows()[0]
     if len(ybrace) == 0:
-        return (ALLL.rows()[0].norm())**2 / c1
+        return v.dot_product(v) / c1
     else:
         sigma = ybrace[len(ybrace)-1]
-        return ((ALLL.rows()[0].norm())**2 * sigma) / c1
+        return v.dot_product(v) * sigma / c1
 
 
 def reduction_step_complex_case(place, B0, list_of_gens, torsion_gen, c13):

src/sage/rings/number_field/galois_group.py

@@ -45,9 +45,11 @@ class GaloisGroup_v1(SageObject):
 
     EXAMPLES::
 
+        sage: # needs sage.symbolic
         sage: from sage.rings.number_field.galois_group import GaloisGroup_v1
         sage: K = QQ[2^(1/3)]
-        sage: G = GaloisGroup_v1(K.absolute_polynomial().galois_group(pari_group=True), K); G
+        sage: pK = K.absolute_polynomial()
+        sage: G = GaloisGroup_v1(pK.galois_group(pari_group=True), K); G
         ...DeprecationWarning: GaloisGroup_v1 is deprecated; please use GaloisGroup_v2
         See https://github.com/sagemath/sage/issues/28782 for details.
         Galois group PARI group [6, -1, 2, "S3"] of degree 3 of the
@@ -96,6 +98,8 @@ def __eq__(self, other):
             sage: G = GaloisGroup_v1(K.absolute_polynomial().galois_group(pari_group=True), K)
             ...DeprecationWarning: GaloisGroup_v1 is deprecated; please use GaloisGroup_v2
             See https://github.com/sagemath/sage/issues/28782 for details.
+
+            sage: # needs sage.symbolic
             sage: L = QQ[sqrt(2)]
             sage: H = GaloisGroup_v1(L.absolute_polynomial().galois_group(pari_group=True), L)
             sage: H == G
@@ -125,6 +129,8 @@ def __ne__(self, other):
             sage: G = GaloisGroup_v1(K.absolute_polynomial().galois_group(pari_group=True), K)
             ...DeprecationWarning: GaloisGroup_v1 is deprecated; please use GaloisGroup_v2
             See https://github.com/sagemath/sage/issues/28782 for details.
+
+            sage: # needs sage.symbolic
             sage: L = QQ[sqrt(2)]
             sage: H = GaloisGroup_v1(L.absolute_polynomial().galois_group(pari_group=True), L)
             sage: H != G

src/sage/rings/number_field/homset.py

@@ -342,10 +342,10 @@ def _element_constructor_(self, x, base_map=None, base_hom=None, check=True):
         are only approximate::
 
             sage: K.<a> = QuadraticField(-7)
-            sage: f = K.hom([CC(sqrt(-7))], check=False)
+            sage: f = K.hom([CC(sqrt(-7))], check=False)                                # needs sage.symbolic
             sage: x = polygen(K)
             sage: L.<b> = K.extension(x^2 - a - 5)
-            sage: L.Hom(CC)(f(a + 5).sqrt(), f, check=False)
+            sage: L.Hom(CC)(f(a + 5).sqrt(), f, check=False)                            # needs sage.symbolic
             Relative number field morphism:
               From: Number Field in b with defining polynomial x^2 - a - 5 over its base field
               To:   Complex Field with 53 bits of precision

src/sage/rings/number_field/maps.py

@@ -79,7 +79,7 @@ def _repr_type(self):
 
     def is_injective(self):
         r"""
-         EXAMPLES::
+        EXAMPLES::
 
             sage: x = polygen(ZZ, 'x')
             sage: K.<a> = NumberField(x^4 + 3*x + 1)
@@ -91,7 +91,7 @@ def is_injective(self):
 
     def is_surjective(self):
         r"""
-         EXAMPLES::
+        EXAMPLES::
 
             sage: x = polygen(ZZ, 'x')
             sage: K.<a> = NumberField(x^4 + 3*x + 1)

src/sage/rings/number_field/number_field.py

@@ -1,4 +1,4 @@
-# sage.doctest: optional - sage.rings.number_field
+# sage.doctest: needs sage.rings.number_field
 """
 Base class for all number fields
 
@@ -135,13 +135,15 @@ cdef class NumberField(Field):
         else:
             codomain_other = other
 
-        from sage.rings.qqbar import AA
-        if codomain_self is AA and codomain_other is AA:
-            return AA
-
-        from sage.rings.qqbar import QQbar
-        if codomain_self in (AA, QQbar) and codomain_other in (AA, QQbar):
-            return QQbar
+        try:
+            from sage.rings.qqbar import AA, QQbar
+        except ImportError:
+            pass
+        else:
+            if codomain_self is AA and codomain_other is AA:
+                return AA
+            if codomain_self in (AA, QQbar) and codomain_other in (AA, QQbar):
+                return QQbar
 
     def ring_of_integers(self, *args, **kwds):
         r"""

src/sage/rings/number_field/number_field_base.pyx

@@ -21,8 +21,8 @@ cdef class NumberFieldElement_base(FieldElement):
     EXAMPLES::
 
         sage: x = polygen(ZZ, 'x')
-        sage: k.<a> = NumberField(x^3 + x + 1)                                                          # optional - sage.rings.number_field
-        sage: isinstance(a, sage.rings.number_field.number_field_element_base.NumberFieldElement_base)  # optional - sage.rings.number_field
+        sage: k.<a> = NumberField(x^3 + x + 1)                                          # needs sage.rings.number_field
+        sage: isinstance(a, sage.rings.number_field.number_field_element_base.NumberFieldElement_base)                  # needs sage.rings.number_field
         True
 
     By design, there is a unique direct subclass::

src/sage/rings/number_field/number_field_element.pyx

@@ -295,9 +295,9 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
         EXAMPLES::
 
             sage: K.<a> = QuadraticField(-1)
-            sage: (1/3 + a/2)._sympy_()                                                 # optional - sympy
+            sage: (1/3 + a/2)._sympy_()                                                 # needs sympy
             1/3 + I/2
-            sage: type(_)                                                               # optional - sympy
+            sage: type(_)                                                               # needs sympy
             <class 'sympy.core.add.Add'>
         """
         a = self.parent().gen()
@@ -506,10 +506,10 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
 
         Verify embeddings are respected::
 
-            sage: cf6c = CyclotomicField(6, embedding=CDF(exp(-pi*I/3))) ; z6c = cf6c.0
-            sage: cf3(z6c)
+            sage: cf6c = CyclotomicField(6, embedding=CDF(exp(-pi*I/3))); z6c = cf6c.0  # needs sage.symbolic
+            sage: cf3(z6c)                                                              # needs sage.symbolic
             -zeta3
-            sage: cf6c(z3)
+            sage: cf6c(z3)                                                              # needs sage.symbolic
             -zeta6
 
         AUTHOR:
@@ -986,9 +986,10 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
             sage: (-a-2).sign()
             -1
 
+            sage: # needs sage.symbolic
             sage: x = polygen(ZZ, 'x')
-            sage: K.<b> = NumberField(x^2 + 2*x + 7, 'b', embedding=CC(-1,-sqrt(6)))    # optional - sage.symbolic
-            sage: b.sign()                                                              # optional - sage.symbolic
+            sage: K.<b> = NumberField(x^2 + 2*x + 7, 'b', embedding=CC(-1,-sqrt(6)))
+            sage: b.sign()
             Traceback (most recent call last):
             ...
             ValueError: a complex number has no sign!
@@ -1822,6 +1823,7 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
             0
             sage: (a + 1/2).real()
             1/2
+            sage: x = polygen(ZZ, 'x')
             sage: K.<a> = NumberField(x^2 + x + 1)
             sage: a.real()
             -1/2
@@ -1865,7 +1867,7 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
             1/2*sqrt3
             sage: a.real()
             -1/2
-            sage: SR(a)                                                                 # optional - sage.symbolic
+            sage: SR(a)                                                                 # needs sage.symbolic
             1/2*I*sqrt(3) - 1/2
             sage: bool(QQbar(I)*QQbar(a.imag()) + QQbar(a.real()) == QQbar(a))
             True
@@ -2214,7 +2216,7 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
 
             sage: x = polygen(ZZ, 'x')
             sage: K.<a> = NumberField(x^2 + 1, embedding=CDF.gen())
-            sage: abs(a+1)
+            sage: abs(a+1)                                                              # needs sage.symbolic
             sqrt(2)
         """
         if mpz_sgn(self.D.value) == 1:
@@ -2506,11 +2508,11 @@ cdef class NumberFieldElement_gaussian(NumberFieldElement_quadratic_sqrt):
         r"""
         EXAMPLES::
 
-            sage: SR(1 + 2*i)                                                           # optional - sage.symbolic
+            sage: SR(1 + 2*i)                                                           # needs sage.symbolic
             2*I + 1
 
             sage: K.<mi> = QuadraticField(-1, embedding=CC(0,-1))
-            sage: SR(1 + mi)                                                            # optional - sage.symbolic
+            sage: SR(1 + mi)                                                            # needs sage.symbolic
             -I + 1
         """
         from sage.symbolic.constants import I
@@ -2616,7 +2618,7 @@ cdef class NumberFieldElement_gaussian(NumberFieldElement_quadratic_sqrt):
 
         EXAMPLES::
 
-            sage: I.log()                                                               # optional - sage.symbolic
+            sage: I.log()                                                               # needs sage.symbolic
             1/2*I*pi
         """
         from sage.symbolic.ring import SR

src/sage/rings/number_field/number_field_element_base.pyx

@@ -1,4 +1,4 @@
-# sage.doctest: optional - sage.libs.pari
+# sage.doctest: needs sage.libs.pari sage.rings.number_field
 """
 Number Field Ideals
 

src/sage/rings/number_field/number_field_element_quadratic.pyx

@@ -738,7 +738,8 @@ def ramification_index(self):
             sage: K.ideal(2).ramification_index()
             Traceback (most recent call last):
             ...
-            NotImplementedError: For an ideal in a relative number field you must use relative_ramification_index or absolute_ramification_index as appropriate
+            NotImplementedError: For an ideal in a relative number field you must use
+            relative_ramification_index or absolute_ramification_index as appropriate
         """
         raise NotImplementedError("For an ideal in a relative number field you must use relative_ramification_index or absolute_ramification_index as appropriate")
 

src/sage/rings/number_field/number_field_ideal.py

@@ -480,9 +480,9 @@ def root_from_approx(f, a):
         sage: root_from_approx(x^2 + 1, CC(0))
         -1*I
 
-        sage: root_from_approx(x^2 - 2, sqrt(2))
+        sage: root_from_approx(x^2 - 2, sqrt(2))                                        # needs sage.symbolic
         sqrt(2)
-        sage: root_from_approx(x^2 - 2, sqrt(3))
+        sage: root_from_approx(x^2 - 2, sqrt(3))                                        # needs sage.symbolic
         Traceback (most recent call last):
         ...
         ValueError: sqrt(3) is not a root of x^2 - 2

src/sage/rings/number_field/number_field_ideal_rel.py

@@ -884,6 +884,7 @@ def _convert_non_number_field_element(self, x):
 
         Examples from :trac:`4727`::
 
+            sage: # needs sage.symbolic
             sage: K.<j,b> = QQ[sqrt(-1), sqrt(2)]
             sage: j
             I
@@ -912,7 +913,8 @@ def _convert_non_number_field_element(self, x):
             sage: L(L)
             Traceback (most recent call last):
             ...
-            TypeError: unable to convert Number Field in a0 with defining polynomial x^2 + 1 over its base field to Number Field in a0 with defining polynomial x^2 + 1 over its base field
+            TypeError: unable to convert Number Field in a0 with defining polynomial x^2 + 1 over its base field
+            to Number Field in a0 with defining polynomial x^2 + 1 over its base field
             sage: L in L
             False
 
@@ -1672,7 +1674,7 @@ def _gen_relative(self):
             sage: k.<a> = NumberField(x^2 + 1); k
             Number Field in a with defining polynomial x^2 + 1
             sage: y = polygen(k)
-            sage: m.<b> = k.extension(y^2+3); m
+            sage: m.<b> = k.extension(y^2 + 3); m
             Number Field in b with defining polynomial x^2 + 3 over its base field
             sage: c = m.gen(); c # indirect doctest
             b
@@ -2455,20 +2457,20 @@ def order(self, *gens, **kwds):
 
         EXAMPLES::
 
-            sage: P.<a,b,c> = QQ[2^(1/2), 2^(1/3), 3^(1/2)]
-            sage: R = P.order([a,b,c]); R
+            sage: P.<a,b,c> = QQ[2^(1/2), 2^(1/3), 3^(1/2)]                             # needs sage.symbolic
+            sage: R = P.order([a,b,c]); R                                               # needs sage.symbolic
             Relative Order in Number Field in sqrt2
              with defining polynomial x^2 - 2 over its base field
 
         The base ring of an order in a relative extension is still `\ZZ`.::
 
-            sage: R.base_ring()
+            sage: R.base_ring()                                                         # needs sage.symbolic
             Integer Ring
 
         One must give enough generators to generate a ring of finite index
         in the maximal order::
 
-            sage: P.order([a, b])
+            sage: P.order([a, b])                                                       # needs sage.symbolic
             Traceback (most recent call last):
             ...
             ValueError: the rank of the span of gens is wrong

src/sage/rings/number_field/number_field_morphisms.pyx

@@ -1064,20 +1064,20 @@ def class_number(self, proof=None):
 
         EXAMPLES::
 
-            sage: ZZ[2^(1/3)].class_number()
+            sage: ZZ[2^(1/3)].class_number()                                            # needs sage.symbolic
             1
-            sage: QQ[sqrt(-23)].maximal_order().class_number()
+            sage: QQ[sqrt(-23)].maximal_order().class_number()                          # needs sage.symbolic
             3
-            sage: ZZ[120*sqrt(-23)].class_number()
+            sage: ZZ[120*sqrt(-23)].class_number()                                      # needs sage.symbolic
             288
 
         Note that non-maximal orders are only supported in quadratic fields::
 
-            sage: ZZ[120*sqrt(-23)].class_number()
+            sage: ZZ[120*sqrt(-23)].class_number()                                      # needs sage.symbolic
             288
-            sage: ZZ[100*sqrt(3)].class_number()
+            sage: ZZ[100*sqrt(3)].class_number()                                        # needs sage.symbolic
             4
-            sage: ZZ[11*2^(1/3)].class_number()
+            sage: ZZ[11*2^(1/3)].class_number()                                         # needs sage.symbolic
             Traceback (most recent call last):
             ...
             NotImplementedError: computation of class numbers of non-maximal orders
@@ -1554,7 +1554,7 @@ def _element_constructor_(self, x):
         (see :trac:`10017`)::
 
             sage: x = polygen(QQ)
-            sage: K.<a> = NumberField(x^3-10)
+            sage: K.<a> = NumberField(x^3 - 10)
             sage: ZK = K.ring_of_integers()
             sage: ZK.basis()
             [1/3*a^2 + 1/3*a + 1/3, a, a^2]
@@ -1665,7 +1665,7 @@ def _magma_init_(self, magma):
 
         OUTPUT:
 
-        a MagmaElement, the magma version of this absolute order
+        a :class:`MagmaElement`, the magma version of this absolute order
 
         EXAMPLES::