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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 
${{\dim}_{{{\mathbb{F}}_{p}}}}{{H}^{n}}\left( H,\ {{\mathbb{F}}_{p}} \right)$ .
