Summary: Let \(p\) be a prime number and \((F, v)\) a valued field. In this paper, we find a presentation for the \(p\)-torsion part of the Brauer group \(\text{Br}(F)\), by means of the valuation \(v\). We only assume that \(F\) has a primitive \(p\)th root of the unity and the residue class field has characteristic not equal to \(p\). This result naturally leads to consider valued fields that we call pre-\(p\)-Henselian fields. It concerns valuations compatible with \(R_p\), the \(p\)-radical of the field. To be precise, \(R_p\) is the radical of the skew-symmetric pairing which associates to a pair \((a, b)\) the class of the symbol algebra \((F; a, b)\) in \(\text{Br}\,F\). In our main result, we state that pre-\(p\)-Henselian fields are precisely the fields for which the Galois group of the maximal Galois \(p\)-extension admits a particular decomposition as a free pro-\(p\) product.