关于Brauer $p$ -维在有限生成域扩张下的行为
On the behaviour of Brauer $p$-dimensions under finitely-generated field extensions
摘要：
本文证明了：若$q \in \mathbb P$或$q = 0$，其中$\mathbb P$为素数集，则对所有不整除$q ^{2} - q$的$p \in \mathbb P$，存在Brauer维数为Brd $(E _{q,k}) = k$和无限绝对Brauer维数为abrd $_{p}(E _{q,k})$的特征$q$域$E _{q,k}\colon \ k \in \mathbb N$.这确保了对于每个有限生成的超越扩张$F _{q,k}/E _{q,k}$，Brd $_{p}(F _{q,k}) = \infty $，$p \dagger q ^{2} - q$。我们还证明了每个序列$a _{p}, b _{p}$，$p \in \mathbb P$，满足条件$a _{2} = b _{2}$和$0 \le b _{p} \le a _{p} \le \infty $，等于序列abrd $_{p}(E), {\rm Brd}_{p}(E)$，$p \in \mathbb P$，为域$E$的特征零。
英文摘要：
The present paper shows that if $q \in \mathbb P$ or $q = 0$, where $\mathbb P$ is the set of prime numbers, then there exist characteristic $q$ fields $E _{q,k}\colon \ k \in \mathbb N$, of Brauer dimension Brd$(E _{q,k}) = k$ and infinite absolute Brauer $p$-dimensions abrd$_{p}(E _{q,k})$, for all $p \in \mathbb P$ not dividing $q ^{2} - q$. This ensures that Brd$_{p}(F _{q,k}) = \infty $, $p \dagger q ^{2} - q$, for every finitely-generated transcendental extension $F _{q,k}/E _{q,k}$. We also prove that each sequence $a _{p}, b _{p}$, $p \in \mathbb P$, satisfying the conditions $a _{2} = b _{2}$ and $0 \le b _{p} \le a _{p} \le \infty $, equals the sequence abrd$_{p}(E), {\rm Brd}_{p}(E)$, $p \in \mathbb P$, for a field $E$ of characteristic zero.
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