  We show that the Nakayama automorphism of a Frobenius algebra $R$ over a
field $k$ is independent of the field (Theorem 4). Consequently, the $k$-dual
functor on left $R$-modules and the bimodule isomorphism type of the $k$-dual
of $R$, and hence the question of whether $R$ is a symmetric $k$-algebra, are
independent of $k$. We give a purely ring-theoretic condition that is necessary
and sufficient for a finite-dimensional algebra over an infinite field to be a
symmetric algebra (Theorem 7).
  Key words: Nakayama automorphism, Frobenius algebra, Frobenius ring,
symmetric algebra, dual module, dual functor, bimodule, Brauer Equivalence.
