A Remark on a Theorem of W. E. H. Berwick

Abstract

Abstract. We indicate and fill a gap in a theorem of W. E. H. Berwick concerning the computation of the fundamental units in a semireal biquadratic field.One of the standard methods for finding a pair of fundamental units in algebraic number fields that have two fundamental units is that The purpose of the present note is to indicate and fill a gap in [1] that occurs in the case of a semireal biquadratic field K, i.e., in the case of a field K that is generated over Q by an element /-. \ va + b{m , a,b,m e Z, m > 0, {m í Q, a + b{m>0, a-b{m<Q.For any «Gíwe denote by a', a", ä" its algebraic conjugates (a' is real and a", 5" are complex-conjugates).The result of Berwick [1, Section 7] is conveniently formulated here as follows: Let R be an order of K containing the ring of integers of Q(\m). Then the set E = (e: e unitofR,e > 1, |e'| < 1, |e"| < 1}is a discrete nonempty set. Let ex be the minimum element of E and i > 1 the fundamental unit of Q(vm ). Then, exe[ = ±i or ±1. (A) // e1ej = ±t, then £,, i is a pair of fundamental units in R. (B) // e,ei = +1, then e,, i or e,, -fi is a pair of fundamental units in R, according as Jl £ R or 41 s R.Remark. In [1] only maximal orders (i.e., the rings of integers) of the various fields are considered. However, it is straightforward to see that the arguments in [1] are also valid if the maximal orders are replaced by orders containing the integers of Q(\/m ) (a simple but important fact is that for such orders R of K we have R' = R, where/?' = {a': a <e R}).