Cyclotomic Swan subgroups and primitive roots

Cyclotomic Swan subgroups and primitive roots

Computation of several cyclotomic Swan subgroups

Let Cl(OK [G]) denote the locally free class group, that is the group of stable isomorphism classes of locally free OK [G]-modules, where OK is the ring of algebraic integers in the number field K and G is a finite group. We show how to compute the Swan subgroup, T (OK [G]), of Cl(OK [G]) when K = Q(ζp), ζp a primitive p-th root of unity, G = C2, where p is an odd (rational) prime so that hp = ...

Primitive Roots of Unity and Cyclotomic Polynomials1

We present a formalization of roots of unity, define cyclotomic polynomi-als and demonstrate the relationship between cyclotomic polynomials and unital polynomials. The following proposition is true (1) For every natural number n holds n = 0 or n = 1 or n ≥ 2. The scheme Comp Ind NE concerns a unary predicate P , and states that: For every non empty natural number k holds P [k] provided the fol...

Primitive Roots of Unity and Cyclotomic Polynomials

One can prove the following proposition (1) For every natural number n holds n = 0 or n = 1 or n ­ 2. The scheme Comp Ind NE concerns a unary predicate P, and states that: For every non empty natural number k holds P[k] provided the parameters satisfy the following condition: • For every non empty natural number k such that for every non empty natural number n such that n < k holds P[n] holds P...

Squaring in cyclotomic subgroups

We propose new squaring formulae for cyclotomic subgroups of certain finite fields. Our formulae use a compressed representation of elements having the property that decompression can be performed at a very low cost. The squaring formulae lead to new exponentiation algorithms in cyclotomic subgroups which outperform the fastest previously-known exponentiation algorithms when the exponent has lo...