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Zbl 1101.11047
Johnston, Henri
On the trace map between absolutely abelian number fields of equal conductor. (English)
[J] Acta Arith. 122, No. 1, 63-74 (2006). ISSN 0065-1036; ISSN 1730-6264

Let $L/K$ be an extension of absolutely abelian number fields of equal conductor, $n$. Then the trace $T_{L/K}(O_L)$ is an ideal of $O_K$ . Let $I(L/K)$ denote the norm over $\bbfQ$ of this ideal. If $e= v_2(n)$, $m= n/2^e$ and $\bbfQ^{(m)}$ is the $m$th cyclotomic field, it is shown that $$I(L/K)=\cases 2^{[K\cap \bbfQ^{(m)}:\bbfQ]}= 2^{[K: \bbfQ]/2^{e-2}} &\text{if }L/K\text{ is wildly ramified},\\ 1 &\text{otherwise}.\endcases$$ This improves a result of {\it K. Girstmair} [Acta Arith. 62, No. 4, 383--389 (1992; Zbl 0739.11051)]. Also, the results here are obtained without the use of Leopoldt's theorem. In fact this method leads to the definition an ``adjusted trace map'' which can be used to reduce the proof of Leopoldt's theorem to the cyclotomic case.
[Charles Parry (Blacksburg)]

MSC 2000:: *11R04 Algebraic numbers
11R33 Integral representations related to algebraic numbers
11R18 Cyclotomic extensions
11R20 Other abelian and metabelian extensions

Keywords: trace map; conductor; Abelian number fields

Citations: Zbl 0739.11051

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