  We provide a simple method of constructing isogeny classes of abelian
varieties over certain fields $k$ such that no variety in the isogeny class has
a principal polarization. In particular, given a field $k$, a Galois extension
$\ell$ of $k$ of odd prime degree $p$, and an elliptic curve $E$ over $k$ that
has no complex multiplication over $k$ and that has no $k$-defined
$p$-isogenies to another elliptic curve, we construct a simple
$(p-1)$-dimensional abelian variety $X$ over $k$ such that every polarization
of every abelian variety isogenous to $X$ has degree divisible by $p^2$. We
note that for every odd prime $p$ and every number field $k$, there exist
$\ell$ and $E$ as above. We also provide a general framework for determining
which finite group schemes occur as kernels of polarizations of abelian
varieties in a given isogeny class.
  Our construction was inspired by a similar construction of Silverberg and
Zarhin; their construction requires that the base field $k$ have positive
characteristic and that there be a Galois extension of $k$ with a certain
non-abelian Galois group.
  Note: Theorem 3.2 in this paper is incorrect. The current version of the
paper includes comments explaining the mistake.
