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

Wilson, Stephen M. J.. "Galois module structure of the rings of integers in wildly ramified extensions." Annales de l'institut Fourier 39.3 (1989): 529-551. <http://eudml.org/doc/74840>.

@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 -