eja: simplify is_invertible() for elements.

diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py
index 15ff26ca909cd1fbec4a2f1fc95e69b69f5bd02f..fb92a310c2edaa27cd9b9cab35dd86d908790469 100644 (file)
--- a/mjo/eja/euclidean_jordan_algebra.py
+++ b/mjo/eja/euclidean_jordan_algebra.py
@@ -492,8 +492,36 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
 
             We can't use the superclass method because it relies on
             the algebra being associative.
+
+            ALGORITHM:
+
+            The usual way to do this is to check if the determinant is
+            zero, but we need the characteristic polynomial for the
+            determinant. The minimal polynomial is a lot easier to get,
+            so we use Corollary 2 in Chapter V of Koecher to check
+            whether or not the paren't algebra's zero element is a root
+            of this element's minimal polynomial.
+
+            TESTS:
+
+            The identity element is always invertible::
+
+                sage: set_random_seed()
+                sage: J = random_eja()
+                sage: J.one().is_invertible()
+                True
+
+            The zero element is never invertible::
+
+                sage: set_random_seed()
+                sage: J = random_eja()
+                sage: J.zero().is_invertible()
+                False
+
             """
-            return not self.det().is_zero()
+            zero = self.parent().zero()
+            p = self.minimal_polynomial()
+            return not (p(zero) == zero)
 
 
         def is_nilpotent(self):