Galois module structure of ideals in wildly ramified cyclic extensions of degree <span class="mathjax-formula">$p^2$</span>

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A NNALES DE L ’ ^INSTITUT F ^OURIER

E ^LDER G ^OVE G ^RIFFITH

Galois module structure of ideals in wildly ramified cyclic extensions of degree p

²

Annales de l’institut Fourier, tome 48, n

^o

2 (1998), p. 609-610

<http://www.numdam.org/item?id=AIF_1998__48_2_609_0>

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Ann. Inst. Fourier, Grenoble 48, 2 (1998), 609-610

CORRIGENDUM

GALOIS MODULE STRUCTURE OF IDEALS IN WILDLY RAMIFIED CYCLIC EXTENSIONS OF DEGREE p

²

by Gove Griffith ELDER

(Article paru dans Ie tome 45 (1995), fascicule 3, pp. 625-647)

The author would like to thank Nigel Byott for pointing out the errors in Theorem 1. Base upon the following lemmas the exponents dn b and hi should read:

• Based upon Lemma 8, dr = \(n + (r + l)p&i)/p2'] — max {|'n/p2],

^2,0 - <°o}-

r-l

• Based upon Lemma 7, b = ^ (f(n — (p — j — l)?^)/?2!

j=0,Sj>p-l

- ^(^-h-JW^21)+ E1 (min {\(\^n)-{s,-j)b,)/p\ -

j=0,p-l^Sj>p-2

e^\(n-(p-j-l)pb,)/p^}-^n-(p-j)pb,)/p^)+ E max {0, \^(n)

J=r,Sj>p-l

eo - ^(?7'+rP6l)/P2^}•

r-l

• Based upon Lemma 7, hi = ^ max{0, f(n — (p — j — l)pb^)/

j=O^Sj-l=i

P2] - r(A2,i(n) - (^ - j)b,)/p] + eo} + _El_.(mm {\{\^i{n) - (s, -

M/p-} - eo, \{n - (p - j - l)pb,)/p^} -^{n - (p - j)pb,)/p^) +

T

E max {0,A2,oH - eo - \(n+rpb^)/p2^}.

J=r,Sj=i

A cursory glance at the statement of Theorem 1 might suggest the appearance of many different types of (R^, R^ X'1) in a given ideal. This is

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610 CORRIGENDUM

misleading. Usually only one or two hi is nonzero. In fact, generally the only nonzero hi are hr and /^r+i-

Finally, we note one may use Nakayama's lemma instead of our methods, see 640-644, to prove Step 3, for example [G.G. Elder and M.J. Madan, Galois module structure of integers in wildly ramified Cp x Cp- extensions, Can. J. Math., 49, n°4 (1997), 722-735].

January 1, 1998.

Gove Griffith ELDER,

University of Nebraska at Omaha College of Arts and Sciences Department of Mathematics Omaha, NE 68182-0243 (USA).

elder@unomaha.edu