(An official and an unoffical electronic version are available.)

We prove that for every field *k* and every positive integer *n*,
there exists an absolutely simple *n*-dimensional abelian variety
over *k*.
We also prove an asymptotic result for finite fields:
For every finite field *k* and positive integer
*n*, we let *S(k,n)* denote the fraction of the isogeny classes of
*n*-dimensional abelian varieties over *k* that consist of absolutely
simple ordinary abelian varieties.
Then for every integer *n*, as *q* approaches infinity over the
prime powers, the ratio *S*(**F**_{q},*n*) approaches 1.