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Full-Text Articles in Number Theory
Multiplicative Properties Of Integral Binary Quadratic Forms, A. G. Earnest, Robert W. Fitzgerald
Multiplicative Properties Of Integral Binary Quadratic Forms, A. G. Earnest, Robert W. Fitzgerald
Articles and Preprints
In this paper, the integral binary quadratic forms for which the set of represented values is closed under k-fold products, for even positive integers k, will be characterized. This property will be seen to distinguish the elements of odd order in the form class group of a fixed discriminant. Further, it will be shown that this closure under k-fold products can always be expressed by a k-linear mapping from (Z2)k to Z2. In the case k = 2, this resolves a conjecture of Aicardi and Timorin.
Bass Series For Small Witt Rings, Robert W. Fitzgerald
Bass Series For Small Witt Rings, Robert W. Fitzgerald
Articles and Preprints
No abstract provided.
Factors Of Dickson Polynomials Over Finite Fields, Robert W. Fitzgerald, Joseph L. Yucas
Factors Of Dickson Polynomials Over Finite Fields, Robert W. Fitzgerald, Joseph L. Yucas
Articles and Preprints
We give new descriptions of the factors of Dickson polynomials over finite fields.
Highly Degenerate Quadratic Forms Over Finite Fields Of Characteristic 2, Robert W. Fitzgerald
Highly Degenerate Quadratic Forms Over Finite Fields Of Characteristic 2, Robert W. Fitzgerald
Articles and Preprints
Let K/F be an extension of finite fields of characteristic two. We consider quadratic forms written as the trace of xR(x), where R(x) is a linearized polynomial. We show all quadratic forms can be so written, in an essentially unique way. We classify those R, with coefficients 0 or 1, where the form has a codimension 2 radical. This is applied to maximal Artin-Schreier curves and factorizations of linearized polynomials.
Sums Of Gauss Sums And Weights Of Irreducible Codes, Robert W. Fitzgerald, Joseph L. Yucas
Sums Of Gauss Sums And Weights Of Irreducible Codes, Robert W. Fitzgerald, Joseph L. Yucas
Articles and Preprints
We develop a matrix approach to compute a certain sum of Gauss sums which arises in the study of weights of irreducible codes. A lower bound on the minimum weight of certain irreducible codes is given.
Pencils Of Quadratic Forms Over Finite Fields, Robert W. Fitzgerald, Joseph L. Yucas
Pencils Of Quadratic Forms Over Finite Fields, Robert W. Fitzgerald, Joseph L. Yucas
Articles and Preprints
A formula for the number of common zeros of a non-degenerate pencil of quadratic forms is given. This is applied to pencils which count binary strings with an even number of 1's prescribed distances apart.
Irreducible Polynomials Over Gf(2) With Three Prescribed Coefficients, Robert W. Fitzgerald, Joseph L. Yucas
Irreducible Polynomials Over Gf(2) With Three Prescribed Coefficients, Robert W. Fitzgerald, Joseph L. Yucas
Articles and Preprints
For an odd positive integer n, we determine formulas for the number of irreducible polynomials of degree n over GF(2) in which the coefficients of xn-1, xn-2 and xn-3 are specified in advance. Formulas for the number of elements in GF(2n) with the first three traces specified are also given.
A Characterization Of Primitive Polynomials Over Finite Fields, Robert W. Fitzgerald
A Characterization Of Primitive Polynomials Over Finite Fields, Robert W. Fitzgerald
Articles and Preprints
No abstract provided.
Isotropy And Factorization In Reduced Witt Rings, Robert W. Fitzgerald
Isotropy And Factorization In Reduced Witt Rings, Robert W. Fitzgerald
Articles and Preprints
We consider reduced Witt rings of finite chain length. We show there is a bound, in terms of the chain length and maximal signature, on the dimension of anisotropic, totally indefinite forms. From this we get the ascending chain condition on principal ideals and hence factorization of forms into products of irreducible forms.
Norms Of Sums Of Squares, Robert W. Fitzgerald
Norms Of Sums Of Squares, Robert W. Fitzgerald
Articles and Preprints
For a finite separable extension K/F of fields of characteristic not 2, the norm of a sum of 2n squares in K is a sum of 2n squares in F. We find explicit identities.
Torsion-Free Modules Over Reduced Witt Rings, Robert W. Fitzgerald
Torsion-Free Modules Over Reduced Witt Rings, Robert W. Fitzgerald
Articles and Preprints
We compute the genus class group of a torsion-free module over a reduced Witt ring of finite stability index. This is applied to modules locally isomorphic to odd degree extensions of formally real fields.
Small Extensions Of Witt Rings, Robert W. Fitzgerald
Small Extensions Of Witt Rings, Robert W. Fitzgerald
Articles and Preprints
We consider certain Witt ring extensions S of a noetherian Witt ring R obtained by adding one new generator. The conditions on the new generator are those known to hold when R is the Witt ring of a Field F, S is the Witt ring of a Field K and K/F is an odd degree extension. We show that if R is of elementary type then so is S.
Gorenstein Witt Rings Ii, Robert W. Fitzgerald
Gorenstein Witt Rings Ii, Robert W. Fitzgerald
Articles and Preprints
The abstract Witt rings which are Gorenstein have been classified when the dimension is one and the classification problem for those of dimension zero has been reduced to the case of socle degree three. Here we classify the Gorenstein Witt rings of fields with dimension zero and socle degree three. They are of elementary type.