Mixed degree number field computations

Journal of Number Theory, 2007

We pose the problem of identifying the set K(G, Ω) of Galois number fields with given Galois group G and root discriminant less than the Serre constant Ω ≈ 44.7632. We definitively treat the cases G = A 4 , A 5 , A 6 and S 4 , S 5 , S 6 , finding exactly 59, 78, 5 and 527, 192, 13 fields respectively. We present other fields with Galois group SL 3 (2), A 7 , S 7 , P GL 2 (7), SL 2 (8), ΣL 2 (8), P GL 2 (9), P ΓL 2 (9), P SL 2 (11), and A 2 5 .2, and root discriminant less than Ω. We conjecture that for all but finitely many groups G, the set K(G, Ω) is empty.

2005

The authors present three-point and four-point covers having bad reduction at 2 and 3 only, with Galois group An or Sn for n equal to 9, 10, 12, 18, 28, and 33. By specializing these covers, they obtain number fields ramified at 2 and 3 only, with Galois group An or Sn for n equal to 9, 10, 11, 12, 17, 18, 25, 28, 30, and 33.

LMS Journal of Computation and Mathematics, 2005

The authors present three-point and four-point covers having bad reduction at 2 and 3 only, with Galois group An or Sn for n equal to 9, 10, 12, 18, 28, and 33. By specializing these covers, they obtain number fields ramified at 2 and 3 only, with Galois group An or Sn for n equal to 9, 10, 11, 12, 17, 18, 25, 28, 30, and 33.

Mathematics of Computation, 1999

We give the lists of all non-primitive number fields of degree eight having two, four and six real places of discriminant less than 6688609, 24363884 and 92810082, respectively, in absolute value. For each field in the lists, we give its discriminant, the discriminant of its subfields, a relative polynomial generating the field over one of its subfields and its discriminant, the corresponding polynomial over Q, and the Galois group of its Galois closure.

Mathematics of Computation, 1994

All algebraic number fields F of degree 5 and absolute discriminant less than 2 x 107 (totally real fields), respectively 5 x 106 (other signatures) are determined. We describe the methods which we applied and list significant data.

Journal of Number Theory, 1990

The minimum discriminant of totally real octic algebraic number fields is determined. It is 282,300,416 and belongs to the ray class field over 9(&) of conductor (7+2&):Y=L?(&) f or cC=(7+2~~+(1+~)~~~)/2. There is-up to isomorphy-only one field of that discriminant. The next two smallest discriminant values are 309,593,125 and 324,000,OOO. For each field we present a full system of fundamental units and its class number.

Mathematics of Computation, 1993

With the mixed-type case now completed, all algebraic number fields of degree 4 with absolute discriminant < 106 have been enumerated. Methods from the totally real and totally complex cases were used without major modification. Isomorphism of fields was determined by a method similar to one of Lenstra. The T2 criterion of Pohst was applied to reduce the number of redundant examples.

2004

We describe a general three step method for constructing number fields with Lie-type Galois groups and discriminants factoring into powers of specified primes. The first step involves extremal solutions of the matrix equation ABC = I. The second step involves extremal polynomial solutions of the equation A(x) + B(x) + C(x) = 0. The third step involves integer solutions of the generalized Fermat equation ax p + by q + cz r = 0. We concentrate here on details associated to the third step and give examples where the field discriminants have the form ±2 a 3 b .

1999

Complete lists of number fields, of given degree n and unramified outside a given finite set S of primes, are both of intrinsic interest and useful in some applications. For degrees n ≤ 5 and S = {∞, 2, 3}, the complete lists have appeared previously; there are in total 85 such fields. Here we give the complete list for n = 6 and S = {∞, 2, 3}, finding in particular exactly 398 such fields. We use a three-pronged approach to obtain this classification: an exhaustive computer search, sextic twinning, and class field theory. Also we completely identify the 2-adic and 3-adic completions of all these degree ≤ 6 fields, this information being one of the focal points of interest and essential in applications. There is a considerable literature on the classification of number fields by means of their discriminants. At present for n = 3, 4, 5, 6, 7 there are large tables, available at [B], giving all number fields of degree n and absolute discriminant less than certain bounds. It would also...

Seema

A number field K is a fit field extension of Q. Its degree is [K: Q]. i.e. its dimension as Q-vector a space. An algebraic number is an algebraic integer if it satisfies a ionic polynomial with integer coefficients, equivalently. Its minimal polynomial over Q should have integer coefficients.