##
**Classgroups and Hermitian modules.**
*(English)*
Zbl 0539.12005

Progress in Mathematics, Vol. 48. Boston-Basel-Stuttgart: Birkhäuser. XVII, 226 p. SFr. 54.00; DM 64.00 (1984).

Let L/K be a finite Galois extension of number fields with Galois group G. The ring \({\mathcal O}_ L\) of integers in L is a locally free \({\mathbb{Z}}G\)-module if and only if L/K is only tamely ramified, and the question naturally arises - and in fact goes back to Emmy Noether - under what additional conditions it turns out to be actually free. Exactly this is the topic which is dealed with in the author’s book ”Galois module structure of algebraic integers” (1983; Zbl 0501.12012).

The theory that is developed there essentially consists of three tools, the classgroup \(C\ell({\mathbb{Z}}G)\), the generalized Lagrange resolvent, and the Galois Gauß sums, and its main theorem states that the class of \({\mathcal O}_ L\) in \(C\ell({\mathbb{Z}}G)\) is determined by the values of the root numbers showing up in the functional equation of the Artin L- functions belonging to the symplectic characters of G. However, the converse is not true, that is, the class of \({\mathcal O}_ L\) does not determine the root numbers; the structure of \(C\ell({\mathbb{Z}}G)\) is simply not strong enough to allow a consequence of that sort. It was the great idea of the author to overcome this difficulty by introducing additional structure, namely the Hermitian structure given by the trace form. And in fact, Ph. Cassou-Noguès and M. Taylor [Math. Ann. 263, 251- 261 (1983; Zbl 0494.12010)] have meanwhile proved the triviality of the root numbers when \({\mathcal O}_ L\) is trivial in this new Hermitian classgroup.

The book under review not only reproduces this proof (at least essentially, neither the relation between the resolvents and the Galois Gauß sums nor Taylor’s group logarithm is treated here), but it provides the whole theory of Hermitian classgroups of orders, which existed before in special cases only (K-theorists have worked in this field - for the purpose of algebraic number theory just unramified extensions can be studied using their results, though).

The more general Hermitian theory that is needed here is worked out in the chapters 2 ”Involution algebras and the Hermitian classgroup” and 3 ”Indecomposable involution algebras”; they also contain the basic definition of the discriminant, the value of which is taken in the Hermitian classgroup. Of course, as with ordinary classgroups, this group is represented as a quotient of certain Hom-groups, so that again one needn’t break the proofs into cases, say the semi-simple case into the simple ones, nor has one to assume the given form to be symplectic or orthogonal or unitary: all these types may very well be simultaneously involved.

The application to group rings then is the content of chapter 4 and the applications in arithmetic, as was already hinted at above, is the topic of the last, the 6th chapter.

The theory that is developed there essentially consists of three tools, the classgroup \(C\ell({\mathbb{Z}}G)\), the generalized Lagrange resolvent, and the Galois Gauß sums, and its main theorem states that the class of \({\mathcal O}_ L\) in \(C\ell({\mathbb{Z}}G)\) is determined by the values of the root numbers showing up in the functional equation of the Artin L- functions belonging to the symplectic characters of G. However, the converse is not true, that is, the class of \({\mathcal O}_ L\) does not determine the root numbers; the structure of \(C\ell({\mathbb{Z}}G)\) is simply not strong enough to allow a consequence of that sort. It was the great idea of the author to overcome this difficulty by introducing additional structure, namely the Hermitian structure given by the trace form. And in fact, Ph. Cassou-Noguès and M. Taylor [Math. Ann. 263, 251- 261 (1983; Zbl 0494.12010)] have meanwhile proved the triviality of the root numbers when \({\mathcal O}_ L\) is trivial in this new Hermitian classgroup.

The book under review not only reproduces this proof (at least essentially, neither the relation between the resolvents and the Galois Gauß sums nor Taylor’s group logarithm is treated here), but it provides the whole theory of Hermitian classgroups of orders, which existed before in special cases only (K-theorists have worked in this field - for the purpose of algebraic number theory just unramified extensions can be studied using their results, though).

The more general Hermitian theory that is needed here is worked out in the chapters 2 ”Involution algebras and the Hermitian classgroup” and 3 ”Indecomposable involution algebras”; they also contain the basic definition of the discriminant, the value of which is taken in the Hermitian classgroup. Of course, as with ordinary classgroups, this group is represented as a quotient of certain Hom-groups, so that again one needn’t break the proofs into cases, say the semi-simple case into the simple ones, nor has one to assume the given form to be symplectic or orthogonal or unitary: all these types may very well be simultaneously involved.

The application to group rings then is the content of chapter 4 and the applications in arithmetic, as was already hinted at above, is the topic of the last, the 6th chapter.

Reviewer: J.Ritter

### MSC:

11R33 | Integral representations related to algebraic numbers; Galois module structure of rings of integers |

00-02 | Research exposition (monographs, survey articles) pertaining to mathematics in general |

11E16 | General binary quadratic forms |

16S34 | Group rings |

11R32 | Galois theory |

11S15 | Ramification and extension theory |

11R42 | Zeta functions and \(L\)-functions of number fields |

11L10 | Jacobsthal and Brewer sums; other complete character sums |

11R52 | Quaternion and other division algebras: arithmetic, zeta functions |

16H05 | Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) |

11R70 | \(K\)-theory of global fields |