Tables of octic fields with a quartic subfield
نویسندگان
چکیده
We describe the computation of extended tables of degree 8 fields with a quartic subfield, using class field theory. In particular we find the minimum discriminants for all signatures and for all the possible Galois groups. We also discuss some phenomena and statistics discovered while making the tables, such as the occurrence of 11 non-isomorphic number fields having the same discriminant, or several pairs of non-isomorphic number fields having the same Dedekind zeta function.
منابع مشابه
Resolution of Sign Ambiguities in Jacobi and Jacobsthal Sums
Let p be a prime = 1 (mod 16). We obtain extensions of known congruences involving parameters of bioctic Jacobi sums (modp). These extensions are used to give an elementary proof of an important congruence of Ήasse relating parameters of quartic and octic Jacobi sums (mod p). This proof leads directly to an elementary resolution of sign ambiguities of parameters of certain quartic, octic, and b...
متن کاملComputation of relative class numbers of CM-fields by using Hecke L-functions
We develop an efficient technique for computing values at s = 1 of Hecke L-functions. We apply this technique to the computation of relative class numbers of non-abelian CM-fields N which are abelian extensions of some totally real subfield L. We note that the smaller the degree of L the more efficient our technique is. In particular, our technique is very efficient whenever instead of simply c...
متن کاملThe Reckoning of Certain Quartic and Octic Gauss Sums
In this brief note, we evaluate certain quartic and octic Gauss sums with the use of theorems on fourth and eighth power difference sets. We recall that a subset H of a finite (additive) abelian group G is said to be a difference set of G [5, p. 64] if for some fixed natural number A, every nonzero element of G can be written as a difference of two elements of/fin exactly A ways. Throughout the...
متن کاملConstructing complete tables of quartic fields using Kummer theory
We explain how to construct tables of quartic fields of discriminant less than or equal to a given bound in an efficient manner using Kummer theory, instead of the traditional (and much less efficient) method using the geometry of numbers. As an application, we describe the computation of quartic fields of discriminant up to 107, the corresponding table being available by anonymous ftp.
متن کاملDetermination of All Nonquadratic Imaginary Cyclic Number Fields of 2-power Degrees with Ideal Class Groups of Exponents
We determine all nonquadratic imaginary cyclic number fields K of 2-power degrees with ideal class groups of exponents < 2, i.e., with ideal class groups such that the square of each ideal class is the principal class, i.e., such that the ideal class groups are isomorphic to some (Z/2Z)m , m > 0. There are 38 such number fields: 33 of them are quartic ones (see Theorem 13), 4 of them are octic ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Math. Comput.
دوره 68 شماره
صفحات -
تاریخ انتشار 1999