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Simplification of Nested Radicals
, 1993
"... Radical simplification is an important part of symbolic computation systems. Until now no algorithms were known for the general denesting problem. If the base field contains all roots of unity, then we give necessary and sufficient conditions for a denesting, and our algorithm computes a denesti ..."
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Radical simplification is an important part of symbolic computation systems. Until now no algorithms were known for the general denesting problem. If the base field contains all roots of unity, then we give necessary and sufficient conditions for a denesting, and our algorithm computes a denesting of ff when it exists. If the base field does not contain all roots of unity, then we show how to compute a denesting that is within depth one of optimal over the base field adjoin a single root of unity. Throughout our paper, we choose to represent a primitive l th root of unity by its symbol i l , rather than as a nested radical. The algorithms require computing the splitting field of the minimal polynomial of ff over k, and have exponential running time. In his magic way, Ramanujan observed a number of striking relationships between certain nested radicals: 3 q 3 p 2 \Gamma 1 = 3 q 1=9 \Gamma 3 q 2=9 + 3 q 4=9 (1) q 3 p 5 \Gamma 3 p 4 = 1=3( 3 p 2 + 3 p 20 ...
Normal Bases over Finite Fields
, 1993
"... Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to repr ..."
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Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to represent finite fields was noted by Hensel in 1888. With the introduction of optimal normal bases, large finite fields, that can be used in secure and e#cient implementation of several cryptosystems, have recently been realized in hardware. The present thesis studies various theoretical and practical aspects of normal bases in finite fields. We first give some characterizations of normal bases. Then by using linear algebra, we prove that F q n has a basis over F q such that any element in F q represented in this basis generates a normal basis if and only if some groups of coordinates are not simultaneously zero. We show how to construct an irreducible polynomial of degree 2 n with linearly i...
How to Tangle with a Nested Radical
 Math. Intelligencer
, 1993
"... this article, we will briefly present some recent theorems for radical simplification, and the algorithms they lead to. For proofs, and complete presentations, the reader is urged to read the original papers. ..."
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this article, we will briefly present some recent theorems for radical simplification, and the algorithms they lead to. For proofs, and complete presentations, the reader is urged to read the original papers.
Galois Field Library: Reference Manual
, 1998
"... Galois Field Library (GFL) is a portable generalpurpose computational library of functions written in C for working over finite fields. The library provides a comprehensive treatment of operations in prime fields and their arbitrary finite extensions. Application programmers should find this librar ..."
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Galois Field Library (GFL) is a portable generalpurpose computational library of functions written in C for working over finite fields. The library provides a comprehensive treatment of operations in prime fields and their arbitrary finite extensions. Application programmers should find this library useful for developing programs in the areas of publickey cryptography, error control coding and combinatorial design. This technical report is a reference manual of GFL. It provides an exhaustive listing of all the features of the library  namely the new data structures and macro definitions introduced in the header files of the library and the prototypes of all GFL library calls. KEY WORDS: Finite fields Data structures Algorithms Library 1 Introduction Galois Field Library (GFL) is a portable generalpurpose computational library of functions written in C for working over finite fields (also called Galois fields). GFL provides routines for field arithmetic and for manipulation of uni...
theory in the Vorlesungen
, 2009
"... We present a translation of §§160–166 of Dedekind’s Supplement XI to Dirichlet’s Vorlesungen über Zahlentheorie, which contain an investigation of the subfields of C. In particular, Dedekind explores the lattice structure of these subfields, by studying isomorphisms between them. He also indicates h ..."
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We present a translation of §§160–166 of Dedekind’s Supplement XI to Dirichlet’s Vorlesungen über Zahlentheorie, which contain an investigation of the subfields of C. In particular, Dedekind explores the lattice structure of these subfields, by studying isomorphisms between them. He also indicates how his ideas apply to Galois theory. After a brief introduction, we summarize the translated excerpt, emphasizing its Galoistheoretic highlights. We then take issue with Kiernan’s characterization of Dedekind’s work in his extensive survey article on the history of Galois theory; Dedekind has a nearly complete realization of the modern “fundamental theorem of Galois theory ” (for subfields of C), in stark contrast to the picture presented by Kiernan at points. We intend a sequel to this article of an historical and philosophical nature. With that in mind, we have sought to make Dedekind’s text accessible to as wide an audience as possible. Thus we include a fair amount of background and exposition. 1
Multiple Left Regular . . .
, 2000
"... Let p1>>pn 0, and p = det kx pj i kn span of the partial derivatives of i;j=1. Let Mp be the linear p. Then Mp is a graded Snmodule. We prove that it is the direct sum of graded left regular representations of Sn. Speci cally, set j = pj, (n, j), and let (t) be the Hilbert polynomial of the span of ..."
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Let p1>>pn 0, and p = det kx pj i kn span of the partial derivatives of i;j=1. Let Mp be the linear p. Then Mp is a graded Snmodule. We prove that it is the direct sum of graded left regular representations of Sn. Speci cally, set j = pj, (n, j), and let (t) be the Hilbert polynomial of the span of all skew Schur functions s = as varies in. Then the graded Frobenius characteristic of Mp is (t) ~ H1n(x; q; t),amultiple of a Macdonald polynomial. Corresponding results are also given for the span of partial derivatives of an alternant over any complex re ection group. Let (i; j) denote the lattice cell in the i +1st row and j +1st column of the positive quadrant of the plane. If L is a diagram with lattice cells (p1;q1);:::;(pn;qn), we set L = det kx pj i yqj i kn i;j=1, and let ML be the linear span of the partial derivatives of L. The bihomogeneity of L and its alternating nature under the diagonal action of Sn gives ML the structure of a bigraded Snmodule. We give a family of examples and some general conjectures about the bivariate Frobenius characteristic of ML for two dimensional diagrams. Key Words: Macdonald polynomials, representations of the symmetric group, complex re ection groups, lattice diagram polynomials
A Basis for the Y¹ Subspace of Diagonal Harmonic Polynomials
"... The space DHn of Sn diagonal harmonics is the collection of polynomials P (x; y) = P (x 1 ; . . . ; xn ; y 1 ; . . . ; y n ) which satisfy the differential equations P n i=1 @ r x i @ s y i P (x; y) = 0 for all r; s 0 (with r + s ? 0). Computer explorations by Haiman have revealed that DHn ..."
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The space DHn of Sn diagonal harmonics is the collection of polynomials P (x; y) = P (x 1 ; . . . ; xn ; y 1 ; . . . ; y n ) which satisfy the differential equations P n i=1 @ r x i @ s y i P (x; y) = 0 for all r; s 0 (with r + s ? 0). Computer explorations by Haiman have revealed that DHn has a number of remarkable combinatorial properties. In particular DHn is an Sn module whose conjectured representation, graded by degree in y, is a sign twisted version of the action of Sn on the parking function module. This conjecture predicts the character of each of the yhomogeneous subspaces Y j of DHn . The space Y 0 of diagonal harmonics with no y dependence is known in the classical theory. In this article we construct a basis for the subspace Y¹ of diagonal harmonics linear in y. Using this basis we prove that the Y¹ specialization of the Parking Function Conjecture is correct, and we provide a formula for the character of Y¹ graded by degree in x. This last fo...