+ # The element we're going to call "minimal_polynomial()" on.
+ # Either myself, interpreted as an element of a finite-
+ # dimensional algebra, or an element of an associative
+ # subalgebra.
+ elt = None
+
+ if self.parent().is_associative():
+ elt = FiniteDimensionalAlgebraElement(self.parent(), self)
+ else:
+ V = self.span_of_powers()
+ assoc_subalg = self.subalgebra_generated_by()
+ # Mis-design warning: the basis used for span_of_powers()
+ # and subalgebra_generated_by() must be the same, and in
+ # the same order!
+ elt = assoc_subalg(V.coordinates(self.vector()))
+
+ # Recursive call, but should work since elt lives in an
+ # associative algebra.
+ return elt.minimal_polynomial()
+
+
+ def span_of_powers(self):
+ """
+ Return the vector space spanned by successive powers of
+ this element.
+ """
+ # The dimension of the subalgebra can't be greater than
+ # the big algebra, so just put everything into a list
+ # and let span() get rid of the excess.
+ V = self.vector().parent()
+ return V.span( (self**d).vector() for d in xrange(V.dimension()) )
+
+
+ def subalgebra_generated_by(self):
+ """
+ Return the associative subalgebra of the parent EJA generated
+ by this element.
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,10).abs()
+ sage: J = eja_rn(n)
+ sage: x = J.random_element()
+ sage: x.subalgebra_generated_by().is_associative()
+ True
+ sage: J = eja_ln(n)
+ sage: x = J.random_element()
+ sage: x.subalgebra_generated_by().is_associative()
+ True
+
+ Squaring in the subalgebra should be the same thing as
+ squaring in the superalgebra::
+
+ sage: J = eja_ln(5)
+ sage: x = J.random_element()
+ sage: u = x.subalgebra_generated_by().random_element()
+ sage: u.matrix()*u.vector() == (u**2).vector()
+ True
+
+ """
+ # First get the subspace spanned by the powers of myself...