Discriminant of the normal closure of a dihedral quartic field

摘要

It is known [3, eq. (1)] that a dihedral quartic field L can be expressed in the form. where a, b, c are integers with (1) (a, b) squarefree, c squarefree a~2-b~2c≠k~2 or ck~2 for any integer k. Let L denote the normal closure of L, that is, The discriminant of L is given by where s denotes the squarefree part of a2 - b2c, and the value of e(= - 2, 0, 2, 4, 6, is given in Tables A(c = 2 (mod 4)), B (c = 3 (mod 4)), C (c & 5 (mod 8)), and D(csl (mod 8)) of [3]. Tables A, B, C, D comprise 8, 8, 8, 27 cases respectively. Each case is specified by congruence conditions on a, b, and c. For example B4 is the case a = b = 1 (mod 2), c = 3 (mod 4), and in this case e = 8.