Gaussian Periods and Units in Certain Cyclic Fields

Abstract

Abstract.We analyze the property of period-unit integer translation (there exists a Gaussian period n and rational integer c such that n + c is a unit) in simplest quadratic, cubic, and quartic fields of arbitrary conductor. This is an extension of work of E. Lehmer, R. Schoof, and L. C. Washington for prime conductor. We also determine the Gaussian period polynomial for arbitrary conductor.

Cubic Polynomials, Linear Shifts, and Ramanujan Simple Cubics

“…Foster's paper [6] has an excellent review of earlier work on the Shanks polynomials and the simplest cubic fields; he also proved that every degree-three cyclic extension of the rationals is generated by a Shanks polynomial (which implies the same for our RSC); this was done earlier by Kersten and Michaliček [7]. Also, Lehmer [11] and Lazarus [10] have shown that the minimal polynomials for so-called cubic Gaussian periods, when composed with some x − a for a an integer, will equal one of the Shanks polynomials (and thus are related to our RSC's).…”

“…Example 5. In an effort to find more equations like (10), we look at the minimal polynomials for 2 cos π/36 and 2 cos π/42. Both have minimal polynomials of degree 12, and both can be factored down into degree three polynomials by adjoining appropriate square roots to the rationals.…”

Normal integral bases and Gaussian periods in the simplest cubic fields

“…When f is square-free, then it is known that η 0 is a generator of a normal integral basis of L/Q (for example, see [14,Proposition 4.31]). For the simplest cubic field L n in the case L n /Q is tamely ramified, the following has been known by Lehmer [13] (when ∆ n = f Ln is a prime number), Châtelet [4] and Lazarus [11] (when ∆ n = f Ln is square-free). If ∆ n = f Ln and f Ln is square-free, then…”

“…From now on, we assume that L n /Q is tamely ramified. If ∆ n = f Ln , then it is known the relation between roots of f n (X) and gaussian periods η i by Lehmer [13, p. 536], Châtelet [4] and Lazarus [11,Proposition 2.2]. Hence, we consider the case where f Ln is not necessarily equal to ∆ n .…”