MATHEMATICS OF COMPUTATION
Volume 00, Number 0, Pages 000–000
S 0025-5718(XX)0000-0

CORRIGENDUM TO THE PAPER “THE BRUMER-STARKCONJECTURE IN SOME FAMILIES OF EXTENSIONS OF
SPECIFIED DEGREE”
CORNELIUS GREITHER, XAVIER-FRANÇOIS ROBLOT, AND BRETT A. TANGEDAL

Barry Smith has found an error in the statement and proof of Lemma 2.5 inour paper [GRT] (Math. Comp. 73 (2004), 297-315). This Lemma concerns acyclic Galois extension K/E of CM fields of odd prime degree p. Towards theend of the proof, it is claimed that every root of unity in E is a norm from K.
Our reasoning for this has a gap (the local part of the argument does not work at
places where the quadratic extension E/E + is split, where F = E + is the maximal
totally real subfield of E), and the statement can indeed fail, as confirmed by aconcrete example calculated by Barry Smith. We will first state a corrected versionof the Lemma, and then we will explain how the proof of Proposition 2.2 has tobe adapted. Fortunately the statement of this Proposition (please refer to [GRT])
need not be changed at all, and therefore the rest of the paper is unaffected. (Theonly other place where Lemma 2.5 is used is in the proof of Proposition 2.1, butthere ζp is not in K, so the relevant case of the Lemma is case (i) below, where theformula remains the same.)
Lemma 2.5 (revised). Let K/E be as above and let H be the Galois group ofK/E (so H is cyclic of odd prime order p). Then(i) If either ζp 6∈ K, or if ζp ∈ K and every root of unity in E is a norm from K(note that we may restrict ourselves to p-power roots of unity in this norm
statement without changing anything, since [K : E] = p), then we have thesame formula as in the original version of the Lemma, that is−|AH
K | = |AE | · e (K/E).

(ii) In the case complementary to (i), we have the corrected formula−|AH
K | = |AE | · e (K/E)/p.

Proof. For part (i), the old proof works. Part (ii) is exactly the case where thenorm index [µE : NK/E K ∗ ∩ µE ] is not 1 (contrary to what was claimed in the oldversion). But then this index is exactly p since H is of order p. The same argument
as before gives the formula, now with the p in the denominator.
We recall that the case ζp 6∈ K was called Case I in [GRT], and the complementary case ζp ∈ K was Case II. We have to redo the proof of Proposition 2.2 in[GRT] when case (ii) of Lemma 2.5 (revised) applies. For convenience, let us callthis the “bad” case, and for clarity we repeat what it means: ζp ∈ K and thereis some (p-power) root of unity in E that is not a norm from K. The rest of thiscorrigendum is concerned with the proof of Proposition 2.2 in the “bad” case.
As in [GRT], we distinguish three cases.
c 1997 American Mathematical Society

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CORNELIUS GREITHER, XAVIER-FRANÇOIS ROBLOT, AND BRETT A. TANGEDAL(a) Both µp∞ (K) and µp∞ (E) are of order p;(b) Both µp∞ (K) and µp∞ (E) are of order pe > p; and(c) |µp∞ (E)| = pe−1 and |µp∞ (K)| = pe .
But by looking at the Galois action one sees that in case (c) the norm µK → µE is
already onto, so this is not a “bad” case! Moreover, (a) can be subsumed into (b)by also allowing e = 1, so we just do (b) for all e ≥ 1.
As before we have
|AK | = pe+1 · [Zp [ζp ] : (ᾱ)] · [Zp : (θ(E/F ))]and|AE | = pe · [Zp : (θ(E/F ))].
Here α is some element with (σ − 1)α = θ(K/F ). Let us write simply θ for thelatter. The module (σ − 1)AK is an epimorphic image of AK /AHK . Now by part (ii)of Lemma 2.5 (revised), and since the term e− (K/E) is a nontrivial power of p, weH
get that |AHK | is a multiple of |AE |, so AK /AK has order dividing |AK |/|AE |, whichis p[Zp [ζp ] : (ᾱ)]. (In [GRT] it says “smaller than” (i.e. properly dividing) insteadof “dividing”; that is the only change so far.) Therefore (σ − 1)AK is annihilatedby (σ − 1)α (formerly and erroneously by α; the extra (σ − 1) now takes care ofthe factor p), and AK is annihilated by (σ − 1)2 α = (σ − 1)θ. So for any ideal awith class in AK we have a(σ−1)θ = (x) for some anti-unit x. We have to find ae
BS-annihilator for the ideal ap θ .
As (x)N is the unit ideal (recall N = 1 + σ + . . . + σ p−1 ), we know that xN is aroot of unity. But since we are in the “bad” case, this entails that xN is the p-th
power of another root of unity. Hence xp−N is a p-th power.
We may find ξ ∈ Z[σ] with p − N = ξ(σ − 1). Recalling from the setting in[GRT] that N θ = 0, we therefore obtaineap θ = ape−1(p−N )θ

= (xpe−1ξ).

e−1

Let us put y = xp ξ and show that y is pe -abelian over F . By the same arguments
as in [GRT], it suffices for this that y σ−1 is a pe -th power in K. But indeed,e−1e−1y σ−1 = xp ξ(σ−1) = xp (p−N ) , and we already observed that x(p−N ) is a p-thpower. So we have proved that Proposition 2.2 is also correct in the “bad” case,that is, the case where the old version of Lemma 2.5 fails.
This essentially concludes our corrigendum. Let us just add that in a way theargument has become esthetically more appealing, since there is no longer an extra
margin in the verification of the pe -abelian property, in contrast with the end of
the first paragraph in the proof of Proposition 2.2 in [GRT], where we said “Thepunch line is that we still have a margin.” Forget that punch line!
We take this opportunity to mention two very small bits in [GRT] which we alsowould like to modify for the sake of clarity and precision: (1) In the Remark justpreceding Propositon 2.2 (page 305), second line, replace “is identical with” by “isincluded in”. (2) In the statement of Proposition 2.2, read “in particular” insteadof “in other words”.
Acknowledgment: We would like to thank Barry Smith for bringing these issues
to our attention and for some illuminating remarks about the sensitivity of theBrumer-Stark conjecture.

CORRIGENDUM TO “BRUMER-STARK IN FAMILIES”

3References[GRT] Cornelius Greither, Xavier-François Roblot, Brett A. Tangedal, The BrumerStark conjecture in some families of extensions of specified degree, Math. Comp. 73 (2004),
297-315.
Institut für theoretische Informatik und Mathematik, Fakultät für Informatik, Universität der Bundeswehr München, 85577 Neubiberg, F.R.Germany
E-mail address: cornelius.greither@unibw.deUniversité de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43
blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France
E-mail address: roblot@math.univ-lyon1.fr
Department of Mathematics and Statistics, University of North Carolina at Greensboro, Greensboro, NC 27412
E-mail address: batanged@uncg.edu