ideals of non-maximal orders, for quadratic fields in particular

src/doc/en/reference/number_fields/index.rst

@@ -38,6 +38,7 @@ Orders, Ideals, Ideal Classes
    sage/rings/number_field/order
    sage/rings/number_field/number_field_ideal
    sage/rings/number_field/number_field_ideal_rel
+   sage/rings/number_field/order_ideal
    sage/rings/number_field/class_group
    sage/rings/number_field/unit_group
    sage/rings/number_field/S_unit_solver

src/sage/rings/number_field/class_group.py

@@ -32,13 +32,11 @@
     sage: I * I.ideal()   # ideal classes coerce to their representative ideal
     Fractional ideal (4, 1/2*a + 3/2)
 
-    sage: O = K.OK(); O
-    Maximal Order in Number Field in a with defining polynomial x^2 + 23
-    sage: O*(2, 1/2*a + 1/2)
+    sage: K*(2, 1/2*a + 1/2)
     Fractional ideal (2, 1/2*a + 1/2)
-    sage: (O*(2, 1/2*a + 1/2)).is_principal()
+    sage: (K*(2, 1/2*a + 1/2)).is_principal()
     False
-    sage: (O*(2, 1/2*a + 1/2))^3
+    sage: (K*(2, 1/2*a + 1/2))^3
     Fractional ideal (1/2*a - 3/2)
 """
 
@@ -66,7 +64,7 @@ class FractionalIdealClass(AbelianGroupWithValuesElement):
         sage: K.<w> = QuadraticField(-23)
         sage: OK = K.ring_of_integers()
         sage: C = OK.class_group()
-        sage: P2a, P2b = [P for P, e in (2*OK).factor()]
+        sage: P2a, P2b = [P for P, e in (2*K).factor()]
         sage: c = C(P2a); c
         Fractional ideal class (2, 1/2*w - 1/2)
         sage: c.gens()
@@ -208,7 +206,7 @@ def is_principal(self):
             sage: K.<w> = QuadraticField(-23)
             sage: OK = K.ring_of_integers()
             sage: C = OK.class_group()
-            sage: P2a, P2b = [P for P, e in (2*OK).factor()]
+            sage: P2a, P2b = [P for P, e in (2*K).factor()]
             sage: c = C(P2a)
             sage: c.is_principal()
             False
@@ -250,7 +248,7 @@ def ideal(self):
             sage: K.<w> = QuadraticField(-23)
             sage: OK = K.ring_of_integers()
             sage: C = OK.class_group()
-            sage: P2a, P2b = [P for P, e in (2*OK).factor()]
+            sage: P2a, P2b = [P for P, e in (2*K).factor()]
             sage: c = C(P2a); c
             Fractional ideal class (2, 1/2*w - 1/2)
             sage: c.ideal()
@@ -315,7 +313,7 @@ def gens(self):
             sage: K.<w> = QuadraticField(-23)
             sage: OK = K.ring_of_integers()
             sage: C = OK.class_group()
-            sage: P2a, P2b = [P for P, e in (2*OK).factor()]
+            sage: P2a, P2b = [P for P, e in (2*K).factor()]
             sage: c = C(P2a); c
             Fractional ideal class (2, 1/2*w - 1/2)
             sage: c.gens()

src/sage/rings/number_field/number_field.py

@@ -10207,7 +10207,7 @@ def hilbert_symbol(self, a, b, P=None):
             0
             sage: K.hilbert_symbol(a, a + 5, K.ideal(5))
             1
-            sage: K.hilbert_symbol(a + 1, 13, (a+6)*K.maximal_order())
+            sage: K.hilbert_symbol(a + 1, 13, (a+6)*K)
             -1
             sage: [emb1, emb2] = K.embeddings(AA)
             sage: K.hilbert_symbol(a, -1, emb1)
@@ -10239,8 +10239,7 @@ def hilbert_symbol(self, a, b, P=None):
         Primes above 2::
 
             sage: K.<a> = NumberField(x^5 - 23)
-            sage: O = K.maximal_order()
-            sage: p = [p[0] for p in (2*O).factor() if p[0].norm() == 16][0]
+            sage: p = [p[0] for p in (2*K).factor() if p[0].norm() == 16][0]
             sage: K.hilbert_symbol(a, a + 5, p)
             1
             sage: K.hilbert_symbol(a, 2, p)
@@ -10288,8 +10287,7 @@ def hilbert_symbol(self, a, b, P=None):
         `a` and `b` do not have to be integral or coprime::
 
             sage: K.<i> = QuadraticField(-1)
-            sage: O = K.maximal_order()
-            sage: K.hilbert_symbol(1/2, 1/6, 3*O)
+            sage: K.hilbert_symbol(1/2, 1/6, 3*K)
             1
             sage: p = 1 + i
             sage: K.hilbert_symbol(p, p, p)
@@ -12794,7 +12792,6 @@ def _splitting_classes_gens_(K, m, d):
     """
     from sage.groups.abelian_gps.abelian_group import AbelianGroup
 
-    R = K.ring_of_integers()
     Zm = IntegerModRing(m)
     unit_gens = Zm.unit_gens()
     Zmstar = AbelianGroup(len(unit_gens), [x.multiplicative_order() for x in unit_gens])

src/sage/rings/number_field/number_field_element.pyx

@@ -174,11 +174,13 @@ def _inverse_mod_generic(elt, I):
         sage: x = polygen(ZZ, 'x')
         sage: OE.<w> = EquationOrder(x^3 - x + 2)
         sage: from sage.rings.number_field.number_field_element import _inverse_mod_generic
-        sage: _inverse_mod_generic(w, 13*OE)
+        sage: _inverse_mod_generic(w, 13)
         6*w^2 - 6
     """
     from sage.matrix.constructor import matrix
     R = elt.parent()
+    if not R.is_maximal():
+        raise NotImplementedError('not implemented for non-maximal orders')
     I = R.number_field().fractional_ideal(I)
     if not I.is_integral():
         raise ValueError("inverse is only defined modulo integral ideals")
@@ -2097,15 +2099,9 @@ cdef class NumberFieldElement(NumberFieldElement_base):
             sage: R = K.maximal_order()
             sage: R(i+1).gcd(2)
             i + 1
-
-        Non-maximal orders are not supported::
-
             sage: R = K.order(2*i)
             sage: R(1).gcd(R(4*i))
-            Traceback (most recent call last):
-            ...
-            NotImplementedError: gcd() for Order in Number Field in i
-            with defining polynomial x^2 + 1 with i = 1*I is not implemented
+            1
 
         The following field has class number 3, but if the ideal
         ``(self, other)`` happens to be principal, this still works::
@@ -2137,7 +2133,7 @@ cdef class NumberFieldElement(NumberFieldElement_base):
             return R.one()
 
         from .order import is_NumberFieldOrder
-        if not is_NumberFieldOrder(R) or not R.is_maximal():
+        if not is_NumberFieldOrder(R):
             raise NotImplementedError("gcd() for %r is not implemented" % R)
 
         K = R.number_field()
@@ -5354,13 +5350,13 @@ cdef class OrderElement_absolute(NumberFieldElement_absolute):
 
             sage: x = polygen(ZZ, 'x')
             sage: OE.<w> = EquationOrder(x^3 - x + 2)
-            sage: w.inverse_mod(13*OE)
+            sage: w.inverse_mod(13)
             6*w^2 - 6
-            sage: w * (w.inverse_mod(13)) - 1 in 13*OE
+            sage: w * (w.inverse_mod(13)) - 1 in 13*OE.number_field()
             True
             sage: w.inverse_mod(13).parent() == OE
             True
-            sage: w.inverse_mod(2*OE)
+            sage: w.inverse_mod(2)
             Traceback (most recent call last):
             ...
             ZeroDivisionError: w is not invertible modulo Fractional ideal (2)

src/sage/rings/number_field/number_field_ideal.py

@@ -803,7 +803,7 @@ def gens_reduced(self, proof=None):
     def gens_two(self):
         r"""
         Express this ideal using exactly two generators, the first of
-        which is a generator for the intersection of the ideal with `Q`.
+        which is a generator for the intersection of the ideal with `\QQ`.
 
         ALGORITHM: uses PARI's :pari:`idealtwoelt` function, which runs in
         randomized polynomial time and is very fast in practice.
@@ -2444,8 +2444,7 @@ def is_coprime(self, other):
         See :trac:`4536`::
 
             sage: E.<a> = NumberField(x^5 + 7*x^4 + 18*x^2 + x - 3)
-            sage: OE = E.ring_of_integers()
-            sage: i,j,k = [u[0] for u in factor(3*OE)]
+            sage: i,j,k = [u[0] for u in factor(3*E)]
             sage: (i/j).is_coprime(j/k)
             False
             sage: (j/k).is_coprime(j/k)

src/sage/rings/number_field/order.py

@@ -17,7 +17,8 @@
 We compute a basis for an order in a relative extension
 that is generated by 2 elements::
 
-    sage: K.<a,b> = NumberField([x^2 + 1, x^2 - 3]); O = K.order([3*a, 2*b])
+    sage: K.<a,b> = NumberField([x^2 + 1, x^2 - 3])
+    sage: O = K.order([3*a, 2*b])
     sage: O.basis()
     [1, 3*a - 2*b, -6*b*a + 6, 3*a]
 
@@ -493,32 +494,40 @@ def ideal(self, *args, **kwds):
             Fractional ideal (7)
             sage: R = K.order(4*a)
             sage: R.ideal(8)
-            Traceback (most recent call last):
-            ...
-            NotImplementedError: ideals of non-maximal orders are not yet supported
+            doctest:warning ... FutureWarning: ...
+            Ideal (8, 32*a) of Order in Number Field in a with defining polynomial x^2 + 7
 
         This function is called implicitly below::
 
             sage: R = EquationOrder(x^2 + 2, 'a'); R
             Maximal Order in Number Field in a with defining polynomial x^2 + 2
             sage: (3,15)*R
+            doctest:warning ... DeprecationWarning: ...
             Fractional ideal (3)
 
         The zero ideal is handled properly::
 
             sage: R.ideal(0)
             Ideal (0) of Number Field in a with defining polynomial x^2 + 2
         """
-        if not self.is_maximal():
-            raise NotImplementedError("ideals of non-maximal orders not yet supported.")
-        from sage.misc.superseded import deprecation
-        deprecation(34806, 'In the future, constructing an ideal of the ring of '
-                           'integers of a number field will use an implementation '
-                           'compatible with ideals of other (non-maximal) orders, '
-                           'rather than returning an integral fractional ideal of '
-                           'its containing number field. Use .fractional_ideal(), '
-                           'together with an .is_integral() check if desired, to '
-                           'avoid your code breaking with future changes to Sage.')
+        if kwds.get('future', False) or not self.is_maximal():
+            if 'future' in kwds:
+                del kwds['future']
+            from sage.rings.number_field.order_ideal import NumberFieldOrderIdeal
+            return NumberFieldOrderIdeal(self, *args, **kwds)
+        if kwds.get('warn', True):
+            if 'warn' in kwds:
+                del kwds['warn']
+            from sage.misc.superseded import deprecation
+            deprecation(34198, 'In the future, constructing an ideal of the ring of '
+                               'integers of a number field will use an implementation '
+                               'compatible with ideals of other (non-maximal) orders, '
+                               'rather than returning an integral fractional ideal of '
+                               'its containing number field. Use .fractional_ideal(), '
+                               'together with an .is_integral() check if desired, to '
+                               'emulate the current behavior.\nSet warn=0 to silence '
+                               'this warning, and future=1 to activate the upcoming '
+                               'behavior already.')
         I = self.number_field().ideal(*args, **kwds)
         if not I.is_integral():
             raise ValueError("ideal must be integral; use fractional_ideal to create a non-integral ideal.")
@@ -542,7 +551,7 @@ def _coerce_map_from_(self, R):
 
     def __mul__(self, right):
         """
-        Create an ideal in this order using the notation ``Ok*gens``
+        Create an ideal in this order using the syntax ``O * gens``
 
         EXAMPLES::
 
@@ -554,17 +563,16 @@ def __mul__(self, right):
             sage: Ok = k.maximal_order(); Ok
             Maximal Order in Number Field in a with defining polynomial x^2 + 5077
             sage: Ok * (11, a + 7)
+            doctest:warning ... DeprecationWarning: ...
             Fractional ideal (11, a + 7)
             sage: (11, a + 7) * Ok
             Fractional ideal (11, a + 7)
         """
-        if self.is_maximal():
-            return self._K.ideal(right)
-        raise TypeError
+        return self.ideal(right)
 
     def __rmul__(self, left):
         """
-        Create an ideal in this order using the notation ``gens*Ok``.
+        Create an ideal in this order using the syntax ``gens * O``.
 
         EXAMPLES::
 
@@ -580,7 +588,7 @@ def __rmul__(self, left):
             sage: 17*Ok
             Fractional ideal (17)
         """
-        return self * left
+        return self.ideal(left)
 
     def is_field(self, proof=True):
         r"""