On principal ideal testing in totally complex quartic fields and the determination of certain cyclotomic constants

Abstract

Abstract. Let if be any totally complex quartic field. Two algorithms are described for determining whether or not any given ideal in if is principal. One of these algorithms is very efficient in practice, but its complexity is difficult to analyze; the other algorithm is computationally more elaborate but, in this case, a complexity analysis can be provided.These ideas are applied to the problem of determining the cyclotomic numbers of order 5 for a prime p = 1 (mod 5). Given any quadratic (or quintic) nonresidue of p, it is shown that these cyclotomic numbers can be efficiently computed in 0((\ogp)3) binary operations.

On the Infrastructure of the Principal Ideal Class of an Algebraic Number Field of Unit Rank One

“…The latter inequality follows from (4.3) of [8]. By Thus, we see that if we are given the representations <p(pi) and <p(p2) for two minima /Ji,/i2 of 0, we can make the giant step <p(ßi)*<P(ß2); and, by Proposition 3.1(h), we can almost precisely predict the value of(p2(ßi)*(p2(ß2)-This information is now used in ALGORITHM 3.2 (The giant step algorithm) Initialization K <-2c4,Ä" <- [k] Step 1 (Baby steps) where ¿W = (<pf ,<pf).…”

“…As already proved in [8] Step 4 (Test) If i > K, then o is not a principal ideal and we terminate the algorithm. If i < K and 9xi] = <p[k} for some fe € {1,2,3,... ,j}, then a is principal and we terminate the algorithm.…”

“…In general, ip will not be the representation of a minimum of 0; but, we can apply a certain reduction procedure to ip = (a, 6) in order to make it the representation of a minimum. For this purpose, we use one of the algorithms of [17] or Buchmann and Williams [8], [9] to obtain a minimum r? in a.…”

“…Principal Ideal Testing. As already mentioned in [17] and [8], it is possible to modify the previous algorithm in order to produce a principal ideal test. To this end, we introduce the notion of a reduced ideal.…”

“…G Proposition 4.2 provides us with a method for computing a reduced ideal in the ideal class of any given ideal of 0. Algorithms for doing this have been given in [17] and [8]. Step 1 (Computation of the reduced principal ideals)…”