Missing coordinates are in the cyclotomic field
Let k be the cyclotomic field
of p-th or 4-th roots of unity according as p > 2 or p = 2.
Washington, Introduction to cyclotomic fields
, Graduate Texts in Math., vol.
It is also well known that Z[[[zeta].sub.n]] + [[[zeta].sup.-1.sub.n]]] is the ring of integers of the nth real cyclotomic field Q([[zeta].sub.n]] + [[zeta].sup.-1.sub.n]]}.
Liang, On the integral basis of the maximal real subfield of a cyclotomic field, J.
Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, 83, Springer, New York, 1997.
Robertson, Heuristics for class numbers of prime-power real cyclotomic fields
, in High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, Fields Inst.
Miller that his methods for finding principal ideals of real cyclotomic fields in ,  may be valid for Q([[zeta].sub.p-1]) at least some small primes p.
Montgomery, Cyclotomic fields with unique factorization, J.
On the Iwasawa [mu]-invariants of branched [Z.sub.p]-covers Jun Ueki On the ring of integers of real cyclotomic fields
Koji YAMAGATA and Masakazu YAMAGISHI Above two, communicated by Shigefumi MORI, M.J.A.
Washington, Quintic poly-  nomials and real cyclotomic fields
with large class numbers, Math.
Birch, Cyclotomic fields
and Kummer extensions, in Algebraic Number Theory (Proc.