Let be a homomorphism. Prove that kernel of is infinite.
Let be irreducible with degree . If has a root on the unit circle then is even and
Let be positive real numbers. Prove that
We apply the AM – GM inequality, thus:
Hence it suffices to prove that which holds because it is equivalent to .
Let be vectors of – dimensional Euclidean space such that . Prove that there exists a permutation of the integers such that
for each .
Let and suppose that , are linear operators from into satisfying
- Show that for all one has
- Show that there exists such that .
- Using the assumptions we have
Consider the linear operator
acting over all
. It may have at most
different eigenvalues. Assuming that
we get that
has infinitely many different eigenvalues
in view of (i). This is a contradiction.