## Thomas Weigel (19/12/2018)

Málstofa í stærðfræði

### Fyrirlesari: Thomas Weigel, Università di Milano-Bicocca

### Titill: The capitulation kernel and Hilbert’s theorem 94

Staðsetning: VR-II, 158

Tími: Miðvikudagur 11. desember kl. 11.00

### Ágrip:

One of the central theorems in Algebraic Number theory

is the finiteness of The capitulation kernel and Hilbert’s theorem 94.

One of the central theorems in Algebraic Number theory

is the finiteness of the Ideal class group of a number field.

The capitulation kernel k(R/O) is the subgroup of ideal classes which

become principal under an extension of Dedekind domains R/O.

Hilbert’s theorem 94 states that for a finite cyclic Galois extension

L/K of number fields of prime power degree, the order of k(R/O) is divisible

by |L:K|. This fact motivated D. Hilbert to formulate his

Principal ideal conjecture which was proved by P. Furtwängler 30 years later.

In this seminar we show a strong version of Hilbert’s theorem 94, which is based

on an abstract version of Hilbert’s theorem 90.f the Ideal class group of a number field.

The capitulation kernel k(R/O) is the subgroup of ideal classes which

become principal under an extension of Dedekind domains R/O.

Hilbert’s theorem 94 states that for a finite cyclic Galois extension

L/K of number fields of prime power degree, the order of k(R/O) is divisible

by |L:K|. This fact motivated D. Hilbert to formulate his

Principal ideal conjecture which was proved by P. Furtwängler 30 years later.

In this seminar we show a strong version of Hilbert’s theorem 94, which is based

on an abstract version of Hilbert’s theorem 90.