Note on fields of small discriminant

Abstract

for totally real fields of degree (g - 1)/2, where g is a prime >3, the smallest discriminant is g(g-3)I2, achieved for R (cos 27r/g), where R is the field of rational numbers. (This is the maximal real sub-field in the field of the gth roots of unity.) This conjecture was often made implicitly even before its verification [1] by Davenport for g =7 (the case g =5 having been trivial). We disprove this for those g, such as 13, 37, etc., for which (g+1)/2 is also prime. We use relative quadratic fields built on base fields of small discriminant, a device used previously by Mayer [3 ] in his work on biquadratic fields. The counterexamples are constructed by adjoining to the totally real base-field k the quantity A1/2, where y is a totally positive integer