  The systole of a closed Riemannian manifold is the minimal length of a
non-contractible closed loop. We give a uniform lower bound for the systole for
large classes of simple arithmetic locally symmetric orbifolds. We establish
new bounds for the translation length of a semisimple element x in SL_n(R) in
terms of its associated Mahler measure. We use these geometric methods to prove
the existence of extensions of number fields in which fixed sets of primes have
certain prescribed splitting behavior.
