Trac #18036: I.parent() should not be the symbolic ring

As suggested in #7545, this ticket defines the imaginary unit `I`
directly as the generator of `QuadraticField(-1)` instead of wrapping it
in a symbolic expression.

**Why?** To allow it to be used in combination with elements of QQbar,
CC, etc., without coercion forcing the expression to SR. For example,
`1.0 + I` is now an element of CC instead of SR.

**How?** We set `I` in sage.all to the generator of ℚ[i], and deprecate
importing it from `sage.symbolic.I`. The symbolic `I` remains available
from `sage.symbolic.constants` for library code working with symbolic
expressions, and as `SR(I)` or `SR.I()`. We create a dedicated subclass
of quadratic number field elements to make it possible to support
features similar to those of symbolic expressions of the form `a + I*b`
that would not make sense for number field elements (or be too hard to
implement, or pollute the namespace).

**Why not ℤ[i]?** Because the class hierarchy of number field and order
elements makes it difficult to provide the compatibility features
mentioned above for elements of both ℤ[i] and ℚ[i]. Having `I` be an
element of ℚ[i] covers almost all use cases (all except working with
algebraic integers?), and people who work with orders are sophisticated
enough to explicitly ask for I ∈ ℤ[i] when they need that. (This is a
debatable choice. We could probably do without the dedicated subclass
for elements of ℚ[i], at the price of breaking backward compatibility a
bit more.)

URL: https://trac.sagemath.org/18036
Reported by: vdelecroix
Ticket author(s): Marc Mezzarobba
Reviewer(s): Vincent Delecroix

src/doc/en/prep/Programming.rst

@@ -728,7 +728,7 @@ not have :math:`I=\sqrt{-1}`, decimal points, or division.
     sage: parent(c)
     Real Field with 53 bits of precision
     sage: parent(d)
-    Symbolic Ring
+    Number Field in I with defining polynomial x^2 + 1 with I = 1*I
     sage: parent(e)
-    Symbolic Ring
+    Complex Field with 53 bits of precision
 

src/sage/all.py

@@ -224,7 +224,6 @@
 from sage.rings.qqbar import _init_qqbar
 _init_qqbar()
 
-
 ###########################################################
 #### WARNING:
 # DO *not* import numpy / matplotlib / networkx here!!
@@ -241,6 +240,8 @@
 true = True
 false = False
 oo = infinity
+from sage.rings.imaginary_unit import I
+i = I
 
 from sage.misc.copying import license
 copying = license

src/sage/categories/rings.py

@@ -999,7 +999,7 @@ def __getitem__(self, arg):
             and orders in number fields::
 
                 sage: ZZ[I]
-                Order in Number Field in I with defining polynomial x^2 + 1 with I = 1*I
+                Order in Number Field in I0 with defining polynomial x^2 + 1 with I0 = 1*I
                 sage: ZZ[sqrt(5)]
                 Order in Number Field in sqrt5 with defining polynomial x^2 - 5 with sqrt5 = 2.236067977499790?
                 sage: ZZ[sqrt(2)+sqrt(3)]
@@ -1054,8 +1054,6 @@ def __getitem__(self, arg):
 
             Embeddings::
 
-                sage: QQ[I](I.pyobject())
-                I
                 sage: a = 10^100; expr = (2*a + sqrt(2))/(2*a^2-1)
                 sage: QQ[expr].coerce_embedding() is None
                 False

src/sage/functions/special.py

@@ -166,7 +166,7 @@
 from sage.symbolic.function import BuiltinFunction
 from sage.libs.mpmath import utils as mpmath_utils
 from sage.functions.all import sqrt, sin, cot, exp
-from sage.symbolic.all import I
+from sage.symbolic.constants import I
 
 
 class SphericalHarmonic(BuiltinFunction):

src/sage/geometry/hyperbolic_space/hyperbolic_coercion.py

@@ -20,7 +20,7 @@
 #***********************************************************************
 
 from sage.categories.morphism import Morphism
-from sage.symbolic.all import I
+from sage.symbolic.constants import I
 from sage.matrix.constructor import matrix
 from sage.modules.free_module_element import vector
 from sage.rings.integer import Integer

src/sage/geometry/hyperbolic_space/hyperbolic_geodesic.py

@@ -83,7 +83,7 @@
 # **********************************************************************
 
 from sage.structure.sage_object import SageObject
-from sage.symbolic.all import I
+from sage.symbolic.constants import I
 from sage.misc.lazy_attribute import lazy_attribute
 from sage.rings.infinity import infinity
 from sage.rings.all import CC, RR

src/sage/geometry/hyperbolic_space/hyperbolic_model.py

@@ -85,7 +85,7 @@
 from sage.functions.all import arccosh
 from sage.rings.all import CC, RR, RDF
 from sage.rings.infinity import infinity
-from sage.symbolic.all import I
+from sage.symbolic.constants import I
 from sage.matrix.constructor import matrix
 from sage.categories.homset import Hom
 

src/sage/geometry/hyperbolic_space/hyperbolic_point.py

@@ -62,7 +62,7 @@
 
 from sage.structure.element import Element
 from sage.structure.richcmp import richcmp, op_NE
-from sage.symbolic.all import I
+from sage.symbolic.constants import I
 from sage.misc.latex import latex
 from sage.structure.element import is_Matrix
 from sage.matrix.constructor import matrix

src/sage/groups/abelian_gps/values.py

@@ -164,7 +164,7 @@ class AbelianGroupWithValuesEmbedding(Morphism):
         sage: embedding = Z4.values_embedding();  embedding
         Generic morphism:
           From: Multiplicative Abelian group isomorphic to C4
-          To:   Symbolic Ring
+          To:   Number Field in I with defining polynomial x^2 + 1 with I = 1*I
         sage: embedding(1)
         1
         sage: embedding(g)
@@ -184,7 +184,7 @@ def __init__(self, domain, codomain):
             sage: AbelianGroupWithValuesEmbedding(Z4, Z4.values_group())
             Generic morphism:
               From: Multiplicative Abelian group isomorphic to C4
-              To:   Symbolic Ring
+              To:   Number Field in I with defining polynomial x^2 + 1 with I = 1*I
         """
         assert domain.values_group() is codomain
         from sage.categories.homset import Hom
@@ -479,7 +479,7 @@ def values_group(self):
 
             sage: Z4 = AbelianGroupWithValues([I], [4])
             sage: Z4.values_group()
-            Symbolic Ring
+            Number Field in I with defining polynomial x^2 + 1 with I = 1*I
         """
         return self._values_group
 
@@ -497,6 +497,6 @@ def values_embedding(self):
             sage: Z4.values_embedding()
             Generic morphism:
               From: Multiplicative Abelian group isomorphic to C4
-              To:   Symbolic Ring
+              To:   Number Field in I with defining polynomial x^2 + 1 with I = 1*I
         """
         return AbelianGroupWithValuesEmbedding(self, self.values_group())

src/sage/interfaces/fricas.py

@@ -1563,8 +1563,7 @@ def _sage_expression(fricas_InputForm):
             D[0, 0, 1](f)(x + y, y)
         """
         from sage.libs.pynac.pynac import register_symbol
-        from sage.symbolic.all import I
-        from sage.symbolic.constants import e, pi
+        from sage.symbolic.constants import e, pi, I
         from sage.calculus.functional import diff
         from sage.functions.log import dilog, lambert_w
         from sage.functions.trig import sin, cos, tan, cot, sec, csc
@@ -1838,10 +1837,9 @@ def _sage_(self):
             <BLANKLINE>
                Cannot convert the value from type Any to InputForm .
         """
-        from sage.rings.all import PolynomialRing, RDF
+        from sage.rings.all import PolynomialRing, RDF, I
         from sage.rings.real_mpfr import RealField
         from sage.symbolic.ring import SR
-        from sage.symbolic.all import I
         from sage.matrix.constructor import matrix
         from sage.modules.free_module_element import vector
         from sage.structure.factorization import Factorization

src/sage/interfaces/gp.py

@@ -891,9 +891,9 @@ def _sage_(self):
 
         EXAMPLES::
 
-            sage: gp(I).sage()
+            sage: gp(SR(I)).sage()
             i
-            sage: gp(I).sage().parent()
+            sage: gp(SR(I)).sage().parent()
             Number Field in i with defining polynomial x^2 + 1 with i = 1*I
 
         ::
@@ -971,7 +971,7 @@ def _complex_mpfr_field_(self, CC):
 
         EXAMPLES::
 
-            sage: z = gp(1+15*I); z
+            sage: z = gp(SR(1+15*I)); z
             1 + 15*I
             sage: z._complex_mpfr_field_(CC)
             1.00000000000000 + 15.0000000000000*I

src/sage/libs/pari/tests.py

@@ -542,10 +542,14 @@
     0
     sage: pari(-1/2).sign()
     -1
-    sage: pari(I).sign()
+    sage: pari(SR(I)).sign()
     Traceback (most recent call last):
     ...
     PariError: incorrect type in gsigne (t_COMPLEX)
+    sage: pari(I).sign()
+    Traceback (most recent call last):
+    ...
+    PariError: incorrect type in gsigne (t_POLMOD)
 
     sage: y = pari('y')
     sage: x = pari('9') + y - y
@@ -1197,7 +1201,7 @@
     [0, 1/2, 0, -3/4, 0, 2, -3/2, 0, -9/16, 40, -116, 117/4, 256000/117, Vecsmall([1]), [Vecsmall([64, 1])], [0, 0, 0, 0, 0, 0, 0, 0]]
     sage: pari([0,0.5,0,-0.75,0]).ellinit()
     [0, 0.500000000000000, 0, -0.750000000000000, 0, 2.00000000000000, -1.50000000000000, 0, -0.562500000000000, 40.0000000000000, -116.000000000000, 29.2500000000000, 2188.03418803419, Vecsmall([0]), [Vecsmall([64, 1])], [0, 0, 0, 0]]
-    sage: pari([0,I,0,1,0]).ellinit()
+    sage: pari([0,SR(I),0,1,0]).ellinit()
     [0, I, 0, 1, 0, 4*I, 2, 0, -1, -64, 352*I, -80, 16384/5, Vecsmall([0]), [Vecsmall([64, 0])], [0, 0, 0, 0]]
     sage: x = SR.symbol('x')
     sage: pari([0,x,0,2*x,1]).ellinit()
@@ -1336,9 +1340,9 @@
     sage: e = pari([0,1,1,-2,0]).ellinit()
     sage: e.ellordinate(0)
     [0, -1]
-    sage: e.ellordinate(I)
+    sage: e.ellordinate(SR(I))
     [0.582203589721741 - 1.38606082464177*I, -1.58220358972174 + 1.38606082464177*I]
-    sage: e.ellordinate(I, precision=128)[0].sage()
+    sage: e.ellordinate(SR(I), precision=128)[0].sage()
     0.58220358972174117723338947874993600727 - 1.3860608246417697185311834209833653345*I
     sage: e.ellordinate(1+3*5^1+O(5^3))
     [4*5 + 5^2 + O(5^3), 4 + 3*5^2 + O(5^3)]
@@ -1361,9 +1365,9 @@
     [0]
     sage: e.ellmul(p, 2)
     [1/4, -7/8]
-    sage: q = e.ellmul(p, 1+I); q
+    sage: q = e.ellmul(p, SR(1+I)); q
     [-2*I, 1 + I]
-    sage: e.ellmul(q, 1-I)
+    sage: e.ellmul(q, SR(1-I))
     [1/4, -7/8]
     sage: for D in [-7, -8, -11, -12, -16, -19, -27, -28]:  # long time (1s)
     ....:     hcpol = hilbert_class_polynomial(D)
@@ -1416,15 +1420,15 @@
     sage: e.ellztopoint(0)
     [0]
 
-    sage: pari(I).ellj()
+    sage: pari(SR(I)).ellj()
     1728.00000000000
-    sage: pari(3*I).ellj()
+    sage: pari(SR(3*I)).ellj()
     153553679.396729
     sage: pari('quadgen(-3)').ellj()
     0.E-54
     sage: pari('quadgen(-7)').ellj(precision=256).sage()
     -3375.000000000000000000000000000000000000000000000000000000000000000000000000
-    sage: pari(-I).ellj()
+    sage: pari(SR(-I)).ellj()
     Traceback (most recent call last):
     ...
     PariError: domain error in modular function: Im(argument) <= 0

src/sage/libs/pynac/pynac.pyx

@@ -1174,9 +1174,9 @@ def py_is_crational_for_doctest(x):
         True
         sage: py_is_crational_for_doctest(1.5)
         False
-        sage: py_is_crational_for_doctest(I.pyobject())
+        sage: py_is_crational_for_doctest(I)
         True
-        sage: py_is_crational_for_doctest(I.pyobject()+1/2)
+        sage: py_is_crational_for_doctest(I+1/2)
         True
     """
     return py_is_crational(x)
@@ -1310,11 +1310,11 @@ def py_is_cinteger_for_doctest(x):
         sage: from sage.libs.pynac.pynac import py_is_cinteger_for_doctest
         sage: py_is_cinteger_for_doctest(1)
         True
-        sage: py_is_cinteger_for_doctest(I.pyobject())
+        sage: py_is_cinteger_for_doctest(I)
         True
-        sage: py_is_cinteger_for_doctest(I.pyobject() - 3)
+        sage: py_is_cinteger_for_doctest(I - 3)
         True
-        sage: py_is_cinteger_for_doctest(I.pyobject() + 1/2)
+        sage: py_is_cinteger_for_doctest(I + 1/2)
         False
     """
     return py_is_cinteger(x)
@@ -2350,20 +2350,21 @@ def init_pynac_I():
 
     EXAMPLES::
 
-        sage: I
+        sage: from sage.libs.pynac.pynac import I as symbolic_I
+        sage: symbolic_I
         I
-        sage: I^2
+        sage: symbolic_I^2
         -1
 
     Note that conversions to real fields will give TypeErrors::
 
-        sage: float(I)
+        sage: float(symbolic_I)
         Traceback (most recent call last):
         ...
         TypeError: unable to simplify to float approximation
-        sage: gp(I)
+        sage: gp(symbolic_I)
         I
-        sage: RR(I)
+        sage: RR(symbolic_I)
         Traceback (most recent call last):
         ...
         TypeError: unable to convert '1.00000000000000*I' to a real number
@@ -2372,53 +2373,53 @@ def init_pynac_I():
 
         sage: C = ComplexField(200); C
         Complex Field with 200 bits of precision
-        sage: C(I)
+        sage: C(symbolic_I)
         1.0000000000000000000000000000000000000000000000000000000000*I
-        sage: I._complex_mpfr_field_(ComplexField(53))
+        sage: symbolic_I._complex_mpfr_field_(ComplexField(53))
         1.00000000000000*I
 
-        sage: I._complex_double_(CDF)
+        sage: symbolic_I._complex_double_(CDF)
         1.0*I
-        sage: CDF(I)
+        sage: CDF(symbolic_I)
         1.0*I
 
-        sage: z = I + I; z
+        sage: z = symbolic_I + symbolic_I; z
         2*I
         sage: C(z)
         2.0000000000000000000000000000000000000000000000000000000000*I
-        sage: 1e8*I
+        sage: 1e8*symbolic_I
         1.00000000000000e8*I
 
-        sage: complex(I)
+        sage: complex(symbolic_I)
         1j
 
-        sage: QQbar(I)
+        sage: QQbar(symbolic_I)
         I
 
-        sage: abs(I)
+        sage: abs(symbolic_I)
         1
 
-        sage: I.minpoly()
+        sage: symbolic_I.minpoly()
         x^2 + 1
-        sage: maxima(2*I)
+        sage: maxima(2*symbolic_I)
         2*%i
 
     TESTS:
 
-        sage: repr(I)
+        sage: repr(symbolic_I)
         'I'
-        sage: latex(I)
+        sage: latex(symbolic_I)
         i
 
         sage: sage.libs.pynac.pynac.init_pynac_I()
         sage: type(sage.libs.pynac.pynac.I)
         <type 'sage.symbolic.expression.Expression'>
         sage: type(sage.libs.pynac.pynac.I.pyobject())
-        <type 'sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic'>
+        <type 'sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_gaussian'>
 
     Check that :trac:`10064` is fixed::
 
-        sage: y = I*I*x / x # so y is the expression -1
+        sage: y = symbolic_I*symbolic_I*x / x # so y is the expression -1
         sage: y.is_positive()
         False
         sage: z = -x / x
@@ -2428,9 +2429,8 @@ def init_pynac_I():
         True
     """
     global pynac_I, I
-    from sage.rings.number_field.number_field import QuadraticField
-    K = QuadraticField(-1, 'I', embedding=CC.gen(), latex_name='i')
-    pynac_I = K.gen()
+    from sage.rings.number_field.number_field import GaussianField
+    pynac_I = GaussianField().gen()
     ginac_pyinit_I(pynac_I)
     I = new_Expression_from_GEx(ring.SR, g_I)
 

src/sage/matrix/matrix0.pyx

@@ -2865,7 +2865,7 @@ cdef class Matrix(sage.structure.element.Matrix):
 
         If not, we get an error message::
 
-            sage: a.add_multiple_of_row(1,0,i)
+            sage: a.add_multiple_of_row(1,0,SR.I())
             Traceback (most recent call last):
             ...
             TypeError: Multiplying row by Symbolic Ring element cannot be done over Rational Field, use change_ring or with_added_multiple_of_row instead.
@@ -2949,7 +2949,7 @@ cdef class Matrix(sage.structure.element.Matrix):
 
         If not, we get an error message::
 
-            sage: a.add_multiple_of_column(1,0,i)
+            sage: a.add_multiple_of_column(1,0,SR.I())
             Traceback (most recent call last):
             ...
             TypeError: Multiplying column by Symbolic Ring element cannot be done over Rational Field, use change_ring or with_added_multiple_of_column instead.

src/sage/matrix/special.py

@@ -1233,7 +1233,7 @@ def elementary_matrix(arg0, arg1=None, **kwds):
 
         sage: E = elementary_matrix(4, row1=1, scale=I)
         sage: E.parent()
-        Full MatrixSpace of 4 by 4 dense matrices over Symbolic Ring
+        Full MatrixSpace of 4 by 4 dense matrices over Number Field in I with defining polynomial x^2 + 1 with I = 1*I
 
         sage: E = elementary_matrix(4, row1=1, scale=CDF(I))
         sage: E.parent()