On the integral basis of the maximal real subfield of a cyclotomic field.

TLDR: In this article, it was shown that the maximal real subfield of the n-th cyclotomic field Q(£n) is an integral basis of the principal order of the real number ζη + ε.

Abstract: Let C„ be a primitive n-th root of unity and #=(?(£,, + ζ') be the maximal real subfield of the n-th cyclotomic field Q(£n). It is proved in this paper that &»!_! {l, ς + C, . . ., (ί,, + Ο 2 } is an integral basis of*. Throughout, the following notations will be used : n a positive integer greater than 2, Q the rational number field, ζη a primitive n-th root of unity, φ (n) the Euler φ-function of «, L = (C«) the «-th cyclotomic field, Κ=ζ)(ζη + ζ~) the maximal real subfield of L, D (L), D (K) the absolute field discriminants of L and K respectively, dn = ά(ζη + C) the discriminant of the real number ζη + ζ\" over , ^L/K relative discriminant of L over K, NK/Q absolute norm taken from *over Q, A = {l, ς + C, · . ·, (Cn + C)^\"} the principal order in #generated by ζη + ζή. The following theorem is due to Lehmer [1]. Theorem 1. The discriminant dn ofthe real number ζη + ζ~ J is given by