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I would like to know if any of our German friends can translate that word for me.

Zerlegung is factorisation, isn't it? So what is factorisation automorphism?

This is taken from Deuring's paper “Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins (vierte Mitteilung)”. This is not the run of the mill question, hopefully it is still a valid question.

This is possibly the first time this term appears, although this is part 4 of a series of paper, so I'm not sure if he has used/defined it in the earlier parts.

Wir betrachten den Fall, daß p in $k_1$ prim bleibt, $$\mathbf{p}=\mathbf{P},\quad\mathbf{p}^{\varphi}=\mathbf{P}.$$ $\varphi$ ist dann Zerlegungsautomorphismus von $\mathbf{P}$ über $k$, also auch von $\mathbf{P}_{\Sigma}$ über dem Körper $P$ der rationalen Zahlen, $$\mathbf{p}=\mathbf{P}_{\Sigma}\quad\text{oder}\quad\mathbf{p}^{\varphi}=\mathbf{P}.$$

Danke sehr!

Rodrigo de Azevedo
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BlackAdder
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    Zerlegung could as well mean partitoining ... – Hagen von Eitzen Jul 12 '13 at 13:51
  • But what would a partitioning automorphism mean though? – BlackAdder Jul 12 '13 at 13:51
  • Dunno. Unfortunately I am not aware of Deuring's paper and that automorphism anyway (and this question is the only google hit :) – Hagen von Eitzen Jul 12 '13 at 13:55
  • Haha, that was the reason I turned to SE! – BlackAdder Jul 12 '13 at 14:06
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    I tried googling "Zerlegungsautomorphismus", which looks (to me) more like a German word than "Zerlegungautomorphism", but I got nothing useful. My best guess is that it means an automorphism of some decomposition, i.e., a bijection that sends some decomposition (of something) to itself, but this is only a guess. – Andreas Blass Jul 12 '13 at 14:45
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    If no one answers, maybe provide a quote of a few sentences before the first time it is used in the paper. – GEdgar Jul 12 '13 at 15:45
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    Maybe "decomposition" instead? There is a decomposition group and a decomposition field. If the group is cyclic, then it'd make sense to call its generator a decomposition automorphism. – Jack Schmidt Jul 12 '13 at 16:33

1 Answers1

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I'm fairly sure that Zerlegungsgruppe is commonly translated as decomposition group. It comes about when studying the splitting of prime ideals in a Galois extension of number fields. The Galois group acts (transitively) on the set of primes lying above a given one. The decomposition group of a prime of the bigger field is its stabilizer inside the Galois group.

Given this it stands to reason that Zerlegungsautomorphismus means: an element of the decomposition group, i.e. an automorphism of the bigger field that maps this prime to itself.

A related concept is that of inertia group (Trägheitsgruppe in German) - a subgroup of the decomposition group that induces the trivial automorphism to the residue class field. The hierarchy of ramification groups then resides inside the inertia group. I'm afraid I don't remember what they are called in German.

The same concepts appear in the study of function fields (of transcendence degree one) over a finite field. Dedekind domains and their fields of quotients being the common umbrella. This is apparently Deuring's theme.

Even in English texts the symbol $Z$ (resp. $T$) often stands for the decomposition group (resp. inertia group). I guess this is a tribute to the contributions of German number theorists. We can then identify $Z/T$ as the Galois group of the related extension of residue class fields. In the listed cases the residue class fields are finite, so $Z/T$ is then necessarily cyclic.

Jyrki Lahtonen
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  • It is possible that (as @Jack Schmidt suggested) Zerlegungsautomorphismus has to be a generator of the decomposition group. We need more context and/or an expert to get more details. – Jyrki Lahtonen Jul 12 '13 at 16:44
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    The Galois group of the residue field extension is cyclic, and lifts to a cyclic-group-mod-$T$, yes. Frobenius. – paul garrett Jul 12 '13 at 16:46