Determinants of circulants of prime power order

Let p be a prime > 3. It is shown that no integral circulant of order pk exists with determinant pk+1 . It is also shown that m is the determinant of an integral 9×9 circulant if and only if (m, 3)=l, or m = 0 mod 27. The proof makes use of a criterion which must be satisfied by the difference of two units in the cyclotomic field of level pk .