Quadratic extensions of totally real quintic fields

In this work, we establish lists for each signature of tenth degree number fields containing a totally real quintic subfield and of discriminant less than in absolute value. For each field in the list we give its discriminant, the discriminant of its subfield, a relative polynomial generating the field over one of its subfields, the corresponding polynomial over , and the Galois group of its Galois closure.

We have examined the existence of several non-isomorphic fields with the same discriminants, and also the existence of unramified extensions and cyclic extensions.

     

Factoring high-degree polynomials over with Niederreiter's algorithm on the IBM SP2

A implementation of Niederreiter's algorithm for factoring polynomials over is described. The most time-consuming part of this algorithm, which consists of setting up and solving a certain system of linear equations, is performed in parallel. Once a basis for the solution space is found, all irreducible factors of the polynomial can be extracted by suitable -computations. For this purpose, asymptotically fast polynomial arithmetic algorithms are implemented. These include Karatsuba & Ofman multiplication, Cantor multiplication and Newton inversion. In addition, a new efficient version of the half-gcd algorithm is presented. Sequential run times for the polynomial arithmetic and parallel run times for the factorization are given. A new ``world record' for polynomial factorization over the binary field is set by showing that a pseudo-randomly selected polynomial of degree 300000 can be factored in about 10 hours on 256 nodes of the IBM SP2 at the Cornell Theory Center.

     

Tenth degree number fields with quintic fields having one real place

In this paper, we enumerate all number fields of degree of discriminant smaller than in absolute value containing a quintic field having one real place. For each one of the (resp. found fields of signature (resp. the field discriminant, the quintic field discriminant, a polynomial defining the relative quadratic extension, the corresponding relative discriminant, the corresponding polynomial over , and the Galois group of the Galois closure are given.

In a supplementary section, we give the first coincidence of discriminant of (resp. nonisomorphic fields of signature (resp. .

     

Mixed degree number field computations

We present a method for computing complete lists of number fields in cases where the Galois group, as an abstract group, appears as a Galois group in smaller degree. We apply this method to find the 25 octic fields with Galois group \({{\mathrm{PSL}}}_2(7)\) and smallest absolute discriminant. We carry out a number of related computations, including determining the octic field with Galois group \(2^3{:}{{\mathrm{GL}}}_3(2)\) of smallest absolute discriminant.     

Matrices over orders in algebraic number fields as sums of -th powers

David R. Richman proved that for every integral matrix is a sum of seven -th powers. In this paper, in light of a question proposed earlier by M. Newman for the ring of integers of an algebraic number field, we obtain a discriminant criterion for every matrix over an order of an algebraic number field to be a sum of (seven) -th powers.

     

Some polynomials over and their Galois groups

Examples of polynomials with Galois group over corresponding to every transitive group through degree eight are calculated, constructively demonstrating the existence of an infinity of extensions with each Galois group over through degree eight. The methods used, which for the most part have not appeared in print, are briefly discussed.

     

All double coverings of cyclotomic fields

A double covering of a Galois extension K/F in the sense of [3] is an extension /K of degree ≤2 such that /F is Galois. In this paper we determine explicitly all double coverings of any cyclotomic extension over the rational number field in the complex number field. We get the results mainly by Galois theory and by using and modifying the results and the methods in [2] and [3]. Project 10571097 supported by NSFC     

Computing automorphisms of abelian number fields

Let be an abelian number field of degree . Most algorithms for computing the lattice of subfields of require the computation of all the conjugates of . This is usually achieved by factoring the minimal polynomial of over . In practice, the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size. In this paper we describe a fast probabilistic algorithm for computing the conjugates of , which is based on -adic techniques. Given and a rational prime which does not divide the discriminant of , the algorithm computes the Frobenius automorphism of in time polynomial in the size of and in the size of . By repeatedly applying the algorithm to randomly chosen primes it is possible to compute all the conjugates of .

     

Intersections of two cofinite subfields

Let F be a field finite-dimensional over subfields A and B. It is known from Galois theory that the degree can be infinite. If F is purely inseparable over B and finitely generated over the prime field, this cannot happen; but it can happen when F is not finitely generated. Received: 15 July 2003     

Calculating formula and divisibility for relative class numbers of abelian function fields

In this paper abelian function fields are restricted to the subfields of cyclotomic function fields. For any abelian function field K/k with conductor an irreducible polynomial over a finite field of odd characteristic, we give a calculating formula of the relative divisor class number of K. And using the given calculating formula we obtain a criterion for checking whether or not the relative divisor class number is divisible by the characteristic of k.     

Monomial Bases for

We consider the polynomial algebra as a module over the mod Steenrod algebra, , being an odd prime. We give a minimal set of generators consisting of monomials and characterise all such `monomial bases'.

     

Galois representations and Hecke operators associated with the mod

We prove that any Hecke eigenclass in the mod cohomology of a congruence subgroup of or has attached to it a mod Galois representation such that the characteristic polynomial of a Frobenius element at a prime equals the Hecke polynomial at .

     

Rumely's local global principle for algebraic PC fields over rings

Let be a finite set of rational primes. We denote the maximal Galois extension of in which all totally decompose by . We also denote the fixed field in of elements in the absolute Galois group of by . We denote the ring of integers of a given algebraic extension of by . We also denote the set of all valuations of (resp., which lie over ) by (resp., ). If , then denotes the ring of integers of a Henselization of with respect to . We prove that for almost all , the field satisfies the following local global principle: Let be an affine absolutely irreducible variety defined over . Suppose that for each and for each . Then . We also prove two approximation theorems for .

     

Subquadratic-time factoring of polynomials over finite fields

New probabilistic algorithms are presented for factoring univariate polynomials over finite fields. The algorithms factor a polynomial of degree over a finite field of constant cardinality in time . Previous algorithms required time . The new algorithms rely on fast matrix multiplication techniques. More generally, to factor a polynomial of degree over the finite field with elements, the algorithms use arithmetic operations in .

The new ``baby step/giant step' techniques used in our algorithms also yield new fast practical algorithms at super-quadratic asymptotic running time, and subquadratic-time methods for manipulating normal bases of finite fields.

     

Differents in modular invariant theory

Let be a graded polynomial algebra over a field k, such that each variable is homogeneous of positive degree. No restrictions are made with respect to the field. Let the finite group G act on A by graded algebra automorphisms and denote the subalgebra of invariants by B. In this paper the various "different ideals" of the extension are studied that define the ramification locus. We prove, for example, that the subring of invariants is itself a polynomial ring if and only if the ramification locus is pure of height one. Here the ramification locus is defined by either the Kahler different, the Noether different or the Galois different. As a consequence we prove that the invariant ring is itself a polynomial ring if and only if there are invariants whose Jacobian determinant does not vanish and is of degree δ, where δ is the degree of the Dedekind different. Using this criterion we give a quick proof of Serre's result that if the invariant ring is a polynomial algebra, then the group is generated by generalized reflections.     

Computation of -invariants of real quadratic fields

Let be a real quadratic field and an odd prime number which splits in . In a previous work, the author gave a sufficient condition for the Iwasawa invariant of the cyclotomic -extension of to be zero. The purpose of this paper is to study the case of this result and give new examples of with , by using information on the initial layer of the cyclotomic -extension of .

     

On the number of certain Galois extensions of local fields

In this paper, we will calculate the number of Galois extensions of local fields with Galois group or .

     

Polynomials with PSL(2, 7) as Galois group

We describe a technique for determining the set-transitivity of the Galois group of a polynomial over the rationals. As an application we give a short proof that the polynomial P₇(x) = x⁷ ? 154x + 99 has the simple group PSL(2, 7) of order 168 as its Galois group over the rationals. A similar method is used to prove that the associated splitting field is not that of the polynomial x⁷ ? 7x + 3 given by Trinks [9].     

Deformations of dihedral -group extensions of fields

Given a -Galois extension of number fields we ask whether it is a specialization of a regular -Galois cover of . This is the ``inverse" of the usual use of the Hilbert Irreducibility Theorem in the Inverse Galois problem. We show that for many groups such arithmetic liftings exist by observing that the existence of generic extensions implies the arithmetic lifting property. We explicitly construct generic extensions for dihedral -groups under certain assumptions on the base field . We also show that dihedral groups of order and have generic extensions over any base field with characteristic different from .

     

Minus class groups of the fields of the -th roots of unity

We show that for any prime number the minus class group of the field of the -th roots of unity admits a finite free resolution of length 1 as a module over the ring . Here denotes complex conjugation in . Moreover, for the primes we show that the minus class group is cyclic as a module over this ring. For these primes we also determine the structure of the minus class group.