1 results
Cohomological Dimension and Schreier's Formula in Galois Cohomology
-
- Journal:
- Canadian Mathematical Bulletin / Volume 50 / Issue 4 / 01 December 2007
- Published online by Cambridge University Press:
- 20 November 2018, pp. 588-593
- Print publication:
- 01 December 2007
-
- Article
-
- You have access
- Export citation
-
Let
$p$ be a prime and 
$F$ a field containing a primitive $p$-th root of unity. Then for 
$n\,\in \,\mathbb{N}$, the cohomological dimension of the maximal pro-$p$-quotient 
$G$ of the absolute Galois group of 
$F$ is at most 
$n$ if and only if the corestriction maps 
${{H}^{n}}\left( H,\ {{\mathbb{F}}_{p}} \right)\,\to \,{{H}^{n}}\left( G,\ {{\mathbb{F}}_{p}} \right)$ are surjective for all open subgroups 
$H$ of index $p$. Using this result, we generalize Schreier's formula for 
${{\dim}_{{{\mathbb{F}}_{p}}}}\,{{H}^{1}}\,\left( H,\ {{\mathbb{F}}_{p}} \right)$ to 
${{\dim}_{{{\mathbb{F}}_{p}}}}{{H}^{n}}\left( H,\ {{\mathbb{F}}_{p}} \right)$.
