Cyclotomic equations and square properties in rings

Linear Groups and Square Properties in Rings

In [1] a proof was given of Fermat’s Two-Square Theorem using the group theoretical structure of the classical modular group. This has been extended in many directions and to other square properties in general rings. In particular in [2] a two-square theorem was given for the Gaussian integers in terms of when ii is a quadratic residue. In this note we examine and survey this technique and the ...

Cyclotomic Completions of Polynomial Rings

The main object of study in this paper is the completion Z[q] = lim ←−n Z[q]/((1−q)(1−q) · · · (1−q)) of the polynomial ring Z[q], which arises from the study of a new invariant of integral homology 3spheres with values in Z[q] announced by the author, which unifies all the sl2 Witten-Reshetikhin-Turaev invariants at various roots of unity. We show that any element of Z[q] is uniquely determine...

Exchange rings, exchange equations, and lifting properties

Computing generator in cyclotomic integer rings

The Principal Ideal Problem (resp. Short Principal Ideal Problem), shorten as PIP (resp. SPIP), consists in finding a generator (resp. short generator) of a principal ideal in the ring of integers of a number field. Several lattice-based cryptosystems rely on the presumed hardness of these two problems. In practice, most of them do not use an arbitrary number field but a power-of-two cyclotomic...

Diophantine Equations in Cyclotomic Fields

where p is a given rational prime? It is almost trivial (from the theory of the Gaussian sum or otherwise) that a solution exists with g =p; it is less trivial that a solution also exists when g = p+p+l; but it is not asserted that solutions do not exist for other values of g. While we are unable to give anything like a complete answer to the problem proposed, we can prove something in this dir...