Search results for "Root of unity"
showing 8 items of 8 documents
Zu einem Satz von Isaacs �ber das Casus-Irreducibilis Ph�nomen
2000
Let \(\Omega \) be a field (of characteristic 0). A prime p is called “bose” (naughty) if \(\Omega \) contains all p-th roots of unity. In this paper the theorem is proved: Let K be an admissible subfield of \(\Omega \) (i.e. for each prime p K contains all p-th roots of unity lying in \(\Omega \)), a an algebraic element of \(\Omega /K\) which is contained in a repeated radical extension of K lying in \(\Omega \). Furthermore let the normal hull L of a over K be contained in \(\Omega \). Then all prime divisors of \(\mid L : K \mid \) are naughty (and L is a repeated radical extension of K with naughty prime exponents). This result generalises a theorem of Isaacs [1] who treats the case \(…
Elementary Integration of Superelliptic Integrals
2021
Consider a superelliptic integral $I=\int P/(Q S^{1/k}) dx$ with $\mathbb{K}=\mathbb{Q}(\xi)$, $\xi$ a primitive $k$th root of unity, $P,Q,S\in\mathbb{K}[x]$ and $S$ has simple roots and degree coprime with $k$. Note $d$ the maximum of the degree of $P,Q,S$, $h$ the logarithmic height of the coefficients and $g$ the genus of $y^k-S(x)$. We present an algorithm which solves the elementary integration problem of $I$ generically in $O((kd)^{\omega+2g+1} h^{g+1})$ operations.
Computing generators of the tame kernel of a global function field
2006
Abstract The group K 2 of a curve C over a finite field is equal to the tame kernel of the corresponding function field. We describe two algorithms for computing generators of the tame kernel of a global function field. The first algorithm uses the transfer map and the fact that the l -torsion can easily be described if the ground field contains the l th roots of unity. The second method is inspired by an algorithm of Belabas and Gangl for computing generators of K 2 of the ring of integers in a number field. We finally give the generators of the tame kernel for some elliptic function fields.
Evaluating Multiple Polylogarithm Values at Sixth Roots of Unity up to Weight Six
2017
We evaluate multiple polylogarithm values at sixth roots of unity up to weight six, i.e. of the form $G(a_1,\ldots,a_w;1)$ where the indices $a_i$ are equal to zero or a sixth root of unity, with $a_1\neq 1$. For $w\leq 6$, we present bases of the linear spaces generated by the real and imaginary parts of $G(a_1,\ldots,a_w;1)$ and present a table for expressing them as linear combinations of the elements of the bases.
The $q$-calculus for generic $q$ and $q$ a root of unity
1996
The $q$-calculus for generic $q$ is developed and related to the deformed oscillator of parameter $q^{1/2}$. By passing with care to the limit in which $q$ is a root of unity, one uncovers the full algebraic structure of ${{\cal Z}}_n$-graded fractional supersymmetry and its natural representation.
Geometrical foundations of fractional supersymmetry
1997
A deformed $q$-calculus is developed on the basis of an algebraic structure involving graded brackets. A number operator and left and right shift operators are constructed for this algebra, and the whole structure is related to the algebra of a $q$-deformed boson. The limit of this algebra when $q$ is a $n$-th root of unity is also studied in detail. By means of a chain rule expansion, the left and right derivatives are identified with the charge $Q$ and covariant derivative $D$ encountered in ordinary/fractional supersymmetry and this leads to new results for these operators. A generalized Berezin integral and fractional superspace measure arise as a natural part of our formalism. When $q$…
Analytic results for planar three-loop integrals for massive form factors
2016
We use the method of differential equations to analytically evaluate all planar three-loop Feynman integrals relevant for form factor calculations involving massive particles. Our results for ninety master integrals at general $q^2$ are expressed in terms of multiple polylogarithms, and results for fiftyone master integrals at the threshold $q^2=4m^2$ are expressed in terms of multiple polylogarithms of argument one, with indices equal to zero or to a sixth root of unity.
Irreducible characters taking root of unity values on $p$-singular elements
2010
In this paper we study finite p-solvable groups having irreducible complex characters chi in Irr(G) which take roots of unity values on the p-singular elements of G.