Galois module structure of the rings of integers in wildly ramified extensions

@article{Wilson1989,
abstract = {The main results of this paper may be loosely stated as follows.Theorem.— Let $N$ and $N^\{\prime \}$ be sums of Galois algebras with group $\Gamma $ over algebraic number fields. Suppose that $N$ and $N^\{\prime \}$ have the same dimension $\{\Bbb Q\}$ and that they are identical at their wildly ramified primes. Then (writing $\{\cal O\}_N$ for the maximal order in $N$)\begin\{\}\{\cal O\}\_N\oplus \{\cal O\}\_N\oplus \{\Bbb Z\}\Gamma \cong \_\{\{\Bbb Z\}\Gamma \} \oplus \{\cal O\}^\{\prime \}\_N\oplus \{\cal O\}\_\{N^\{\prime \}\}\oplus \{\Bbb Z\}\Gamma . \end\{\}In many cases $\{\cal O\}_N \cong _\{\{\Bbb Z\}\Gamma \} \{\cal O\}_\{N^\{\prime \}\}.$The role played by the root numbers of $N$ and $N^\{\prime \}$ at the symplectic characters of $\Gamma $ in determining the relationship between the $\{\Bbb Z\}\Gamma $-modules $\{\cal O\}_N$ and $\{\cal O\}_\{N^\{\prime \}\}$ is described. The theorem includes as a special case the theorem of M. J. Taylor on the structure of the ring of integers in a tamely ramified extension and it employs many of the results employed by Taylor in the proof of his theorem.},
author = {Wilson, Stephen M. J.},
journal = {Annales de l'institut Fourier},
keywords = {Galois module structure; root numbers; symplectic characters; ring of integers; wildly ramified extension; resolvents; transfer; Galois-Gauss sums; representation of orders; class group},
language = {eng},
number = {3},
pages = {529-551},
publisher = {Association des Annales de l'Institut Fourier},
title = {Galois module structure of the rings of integers in wildly ramified extensions},
url = {http://eudml.org/doc/74840},
volume = {39},
year = {1989},
}

TY - JOUR
AU - Wilson, Stephen M. J.
TI - Galois module structure of the rings of integers in wildly ramified extensions
JO - Annales de l'institut Fourier
PY - 1989
PB - Association des Annales de l'Institut Fourier
VL - 39
IS - 3
SP - 529
EP - 551
AB - The main results of this paper may be loosely stated as follows.Theorem.— Let $N$ and $N^{\prime }$ be sums of Galois algebras with group $\Gamma $ over algebraic number fields. Suppose that $N$ and $N^{\prime }$ have the same dimension ${\Bbb Q}$ and that they are identical at their wildly ramified primes. Then (writing ${\cal O}_N$ for the maximal order in $N$)\begin{}{\cal O}_N\oplus {\cal O}_N\oplus {\Bbb Z}\Gamma \cong _{{\Bbb Z}\Gamma } \oplus {\cal O}^{\prime }_N\oplus {\cal O}_{N^{\prime }}\oplus {\Bbb Z}\Gamma . \end{}In many cases ${\cal O}_N \cong _{{\Bbb Z}\Gamma } {\cal O}_{N^{\prime }}.$The role played by the root numbers of $N$ and $N^{\prime }$ at the symplectic characters of $\Gamma $ in determining the relationship between the ${\Bbb Z}\Gamma $-modules ${\cal O}_N$ and ${\cal O}_{N^{\prime }}$ is described. The theorem includes as a special case the theorem of M. J. Taylor on the structure of the ring of integers in a tamely ramified extension and it employs many of the results employed by Taylor in the proof of his theorem.
LA - eng
KW - Galois module structure; root numbers; symplectic characters; ring of integers; wildly ramified extension; resolvents; transfer; Galois-Gauss sums; representation of orders; class group
UR - http://eudml.org/doc/74840
ER -