(An unofficial electronic version is available.)

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 *l* 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 *l* 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.
(Silverberg and Zarhin have a
new construction
that works over an arbitrary number field.)