eja: simplify the way we do field extensions.

[sage.d.git] / mjo / eja / eja_algebra.py

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index 3f2127d7df9b811a599ba1da01b7b100ab0a5573..515e25f9c97de0ae102e8dc6bd7b29db89e39dd0 100644 (file)
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -17,11 +17,10 @@ from sage.misc.prandom import choice
 from sage.misc.table import table
 from sage.modules.free_module import FreeModule, VectorSpace
 from sage.rings.integer_ring import ZZ
-from sage.rings.number_field.number_field import NumberField, QuadraticField
+from sage.rings.number_field.number_field import QuadraticField
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
 from sage.rings.rational_field import QQ
 from sage.rings.real_lazy import CLF, RLF
-from sage.structure.element import is_Matrix
 
 from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
 from mjo.eja.eja_utils import _mat2vec
@@ -406,8 +405,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
 
         EXAMPLES:
 
-        Our inner product satisfies the Jordan axiom, which is also
-        referred to as "associativity" for a symmetric bilinear form::
+        Our inner product is "associative," which means the following for
+        a symmetric bilinear form::
 
             sage: set_random_seed()
             sage: J = random_eja()
@@ -911,7 +910,7 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
             z = R.gen()
             p = z**2 - 2
             if p.is_irreducible():
-                field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt())
+                field = field.extension(p, 'sqrt2', embedding=RLF(2).sqrt())
                 basis = tuple( s.change_ring(field) for s in basis )
             self._basis_normalizers = tuple(
                 ~(self.natural_inner_product(s,s).sqrt()) for s in basis )
@@ -940,8 +939,9 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
         else:
             basis = ( (b/n) for (b,n) in izip(self.natural_basis(),
                                               self._basis_normalizers) )
-            field = self.base_ring().base_ring() # yeeeaahhhhhhh
-            J = MatrixEuclideanJordanAlgebra(field,
+
+            # Do this over the rationals and convert back at the end.
+            J = MatrixEuclideanJordanAlgebra(QQ,
                                              basis,
                                              self.rank(),
                                              normalize_basis=False)
@@ -950,7 +950,14 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
             # p might be missing some vars, have to substitute "optionally"
             pairs = izip(x.base_ring().gens(), self._basis_normalizers)
             substitutions = { v: v*c for (v,c) in pairs }
-            return p.subs(substitutions)
+            result = p.subs(substitutions)
+
+            # The result of "subs" can be either a coefficient-ring
+            # element or a polynomial. Gotta handle both cases.
+            if result in QQ:
+                return self.base_ring()(result)
+            else:
+                return result.change_ring(self.base_ring())
 
 
     @staticmethod
@@ -1013,6 +1020,7 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
         Xu = cls.real_unembed(X)
         Yu = cls.real_unembed(Y)
         tr = (Xu*Yu).trace()
+
         if tr in RLF:
             # It's real already.
             return tr
@@ -1067,6 +1075,14 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA):
         sage: e2*e2
         e2
 
+    In theory, our "field" can be any subfield of the reals::
+
+        sage: RealSymmetricEJA(2, AA)
+        Euclidean Jordan algebra of dimension 3 over Algebraic Real Field
+        sage: RealSymmetricEJA(2, RR)
+        Euclidean Jordan algebra of dimension 3 over Real Field with
+        53 bits of precision
+
     TESTS:
 
     The dimension of this algebra is `(n^2 + n) / 2`::
@@ -1256,10 +1272,12 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra):
         if not n.mod(2).is_zero():
             raise ValueError("the matrix 'M' must be a complex embedding")
 
-        field = M.base_ring() # This should already have sqrt2
+        # If "M" was normalized, its base ring might have roots
+        # adjoined and they can stick around after unembedding.
+        field = M.base_ring()
         R = PolynomialRing(field, 'z')
         z = R.gen()
-        F = NumberField(z**2 + 1,'i', embedding=CLF(-1).sqrt())
+        F = field.extension(z**2 + 1, 'i', embedding=CLF(-1).sqrt())
         i = F.gen()
 
         # Go top-left to bottom-right (reading order), converting every
@@ -1398,7 +1416,7 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA):
         """
         R = PolynomialRing(field, 'z')
         z = R.gen()
-        F = NumberField(z**2 + 1, 'I', embedding=CLF(-1).sqrt())
+        F = field.extension(z**2 + 1, 'I')
         I = F.gen()
 
         # This is like the symmetric case, but we need to be careful:
@@ -1533,7 +1551,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra):
         if M.ncols() != n:
             raise ValueError("the matrix 'M' must be square")
         if not n.mod(4).is_zero():
-            raise ValueError("the matrix 'M' must be a complex embedding")
+            raise ValueError("the matrix 'M' must be a quaternion embedding")
 
         # Use the base ring of the matrix to ensure that its entries can be
         # multiplied by elements of the quaternion algebra.
@@ -1705,7 +1723,10 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra,
                     S.append(Sij_J)
                     Sij_K = cls.real_embed(K*Eij - K*Eij.transpose())
                     S.append(Sij_K)
-        return S
+
+        # Since we embedded these, we can drop back to the "field" that we
+        # started with instead of the quaternion algebra "Q".
+        return ( s.change_ring(field) for s in S )
 
 
     def __init__(self, n, field=QQ, **kwargs):