# Third Galois cohomology group of function fields of curves over number fields

## TL;DR

This paper investigates the structure of the third Galois cohomology group of function fields of curves over totally imaginary number fields, establishing that all elements are symbols under certain conditions.

## Contribution

It proves that when the function field contains a primitive p-th root of unity, all elements in the third Galois cohomology group are symbols, advancing understanding of cohomological properties of such fields.

## Key findings

- All elements in the third Galois cohomology group are symbols under the given conditions.
- The result applies specifically to function fields of curves over totally imaginary number fields containing primitive p-th roots of unity.

## Abstract

Let F be the function field of a curve over totally imaginary number field. Let p be a prime. If F contains a primitive p th root of unity, then every element in the third Galois cohomology group of F with values in the group of p th roots of unity, is a symbol.