# Hilbert-symbol equivalence of global function fields

In: Mathematica Slovaca, vol. 51, no. 4
Alfred Czogała

## Details:

Year, pages: 2001, 393 - 401
Hilbert-symbol equivalence of degree $\ell$ between two global fields containing a primitive $\ell$th root of unity is an isomorphism between the groups of $\ell$th power classes of these fields preserving Hilbert symbols of degree $\ell$. The Hilbert-symbol equivalence of degree $\ell$ is said to be tame if it preserves the $\frak p$@-orders modulo $\ell$. In the paper we prove that if $\ell$ is an odd prime number, then any two global function fields are Hilbert equivalent. We find also necessary and sufficient conditions for tame Hilbert-symbol equivalence of global function fields for all prime numbers $\ell ≥ 2$.