On the class-number of the maximal real subfield of a cyclotomic field

Abstract

For any square-free positive integer m, let H(m) be the class-number of the field Q(ς_m+ς_m^-1 ), where ς_m is a primitive m-th root of unity. We show that if m = {3(8g + 5)}² - 2 is a square-free integer, where g is a positive integer, then H(4m) > 1. Similar result holds for a square-free integer m = {3(8g+7)}²- 2, where g is a positive integer. We also show that nH|(4m) for certain positive integers m and n.

Mathematics Subject Classification (2010): 11R29, 11R18, 11R11.

Keywords: Maximal real subeld of cyclotomic field, real quadratic field, class number