EUCLIDEAN MINIMA AND CENTRAL DIVISION ALGEBRAS

4.1 Euclidean Division

We saw in Lecture 3 how to efficiently multiply integers, and, using Kronecker substitution, how to efficiently multiply polynomials with integer coefficients. This gives us what we need to multiply elements in finite fields, provided that we have a way to reduce the result to our standard representations of Fp ' Z/pZ and Fq ' Fp[x]/(f), using integers in [0, p− 1] and polynomials of degree les...

Triangulaizability of Algebras Over Division Rings

Correspondences between Valued Division Algebras and Graded Division Algebras

If D is a tame central division algebra over a Henselian valued field F , then the valuation on D yields an associated graded ring GD which is a graded division ring and is also central and graded simple over GF . After proving some properties of graded central simple algebras over a graded field (including a cohomological characterization of its graded Brauer group), it is proved that the map ...

Division Algebras and Supersymmetry

Supersymmetry is deeply related to division algebras. For example, nonabelian Yang–Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or 10. The same is true for the Green–Schwarz superstring. In both cases, supersymmetry relies on the vanishing of a certain trilinear expression involving a spinor field. The reason for this...

Toral Automorphisms and Applications to Norm-euclidean Domains and Euclidean Minima

A number field modulo its ring of integers is a torus, and the units act on this torus as hyperbolic toral automorphisms. We show sufficient control over the distribution of the rational orbits to give a new, non-computational proof that Q( √ 2 + √ 2) is a normEuclidean number field, and an improved upper bound for the Euclidean minima of real cyclotomic fields of power of 2 conductor.