  Motivated by the discovery that the eighth root of the theta series of the
E_8 lattice and the 24th root of the theta series of the Leech lattice both
have integer coefficients, we investigate the question of when an arbitrary
element f in R (where R = 1 + xZ[[x]]) can be written as f = g^n for g in R, n
>= 2. Let P_n := {g^n : g in R} and let mu_n := n Product_{p|n} p. We show
among other things that (i) for f in R, f in P_n <=> f mod mu_n in P_n, and
(ii) if f in P_n, there is a unique g in P_n with coefficients mod mu_n/n such
that f == g^n (mod mu_n). In particular, if f == 1 (mod mu_n) then f in P_n.
The latter assertion implies that the theta series of any extremal even
unimodular lattice in R^n (e.g. E_8 in R^8) is in P_n if n is of the form 2^i
3^j 5^k (i >= 3). There do not seem to be any exact analogues for codes,
although we show that the weight enumerator of the r-th order Reed-Muller code
of length 2^m is in P_{2^r}. We give a number of other results and conjectures,
and establish a conjecture of Paul D. Hanna that there is a unique element f in
P_n (n != 2) with coefficients restricted to the set {1, 2, ..., n}.
