Talk:Whitehead torsion (Ex)
We can compute that
- so the element is a unit. The map can be described as follows.
There is a ring homomorphism sending a generator to . Now we consider the composite
By definition it takes the class of some automorphism to the real number . The above composite is a well-defined group homomorphism as all single maps are well-defined group homomorphisms, so it remains to check whether it factors over the Whitehead group. So let . Then by left multiplication it defines a map ;
and inducing it up to the complex numbers gives the map ; \[\xymatrix{\Cc \ar[r]^-{e^{\frac{2\pi i\alpha}{5}}} & \Cc}\] given by multiplication by . On the determinant is of course the identity map, so we need to compute ;
which follows since for all real numbers , in particular for .
Now to show that generates an infinite cyclic subgroup of it suffices to show that because then is not a torsion element.
For this we simply calculate that ;
hence it follows that ;
since . In particular applying the logarithm we get ;
which we wanted to show.