Corrigenda and addenda to “Fundamentality of a cubic unit 𝑢 for ℤ[𝕦]”

Abstract

Due to a certain ambiguity present in section 3 of [E. Thomas, Fundamental units for orders in certain cubic number fields, J. Reine Angew. Math. 310 (1979), 33-55], it became necessary to amend a crucial definition and a few proofs appearing in our article Fundamentality of a cubic unit u for Z[u]. Here, the necessary corrections are provided. Our article Fundamentality of a cubic unit u for Z[u], Math. Comp. 80 (2011), 563-578, depends, in a crucial manner, on a theorem due to E. Thomas as referenced in the article. Recently it has come to our notice that there is an ambiguity in E. Thomas's paper which appears to result from a typographical error. Namely, Thomas defines a quantity Δ assuming k := φ(0) = ±1 whereas the proof of his theorem (3.1) needs k = −φ(0) = ±1. At first glance, this might appear inconsequential in view of the fact that replacing φ(X) by −φ(−X) affects neither the order under consideration nor the conclusion of the theorem (3.1); however, a careful study of the proof shows the need to have k = −φ(0) = ±1. This ambiguity affects a part of our definition of the Thomas number. Although, fortunately, the main theorem of our article remains entirely valid, some definitions and computations need to be rectified. Below we list the necessary corrections. Throughout the following, we employ the same notation and terminology as in the article. Likewise, the section numbers appearing below correspond with the respective sections of the article. Recently, S. R. Louboutin has kindly sent us a preprint of his paper entitled "On the fundamental units of a totally real cubic order generated by a unit" in which he provides a new and shorter proof of our main theorem which does not depend on the theorem of E. Thomas mentioned above. 1.1. The definition of θ (a, b) should read as follows for the first two cases: θ(a, b) := |u − v|(1 + w) if {u, v} ⊂ I + , −|u − v|(1 + w) if {u, v} ⊂ I. Likewise, the final "−" should read "+". 4.3. Append "We postpone the special case of n = 3 to be handled separately. From now on, assume n ≥ 4 unless otherwise specified." to the end of the first paragraph. Accordingly, all inequalities and equalities in that subsection comparing n to 3 should have "3" replaced with "4", including those of the form "n = k + 3 ≥ 3". α (m, n) should be defined as n 2 − n + 1 − m /n and β (m, n) should be defined as n + 13/10. For h, the calculations should read: "Both criti