Let \(X\) be an integral scheme and let \(K=k(X)\) be its field of rational functions. An element \(\xi\) in the Witt ring \(W(K)\) is said to be unramified if it belongs to the image of the canonical map \(W({\mathcal O}_{X,x})\rightarrow W(K)\) for every height-one point \(x\in X.\) It is said that purity holds for \(X\) if every unramified element of \(W(K)\) belongs to the image of the canonical map \(W(X)\rightarrow W(K).\) The main result of the paper says that purity holds for \(X\)= Spec\((A)\), where \(A\) is a regular local ring containing a field of characteristic not 2. So far purity has been known to hold for regular integral Noetherian schemes of dimension at most 2 [J.-P. Colliot-Thélène and J.-J. Sansuc, Math. Ann. 244, 105-134 (1979; Zbl 0418.14016)] and for regular integral Noetherian affine schemes of dimension 3 [M. Ojanguren, R. Parimala, R. Sridharan and V. Suresh, Proc. Lond. Math. Soc. (2) 59, 521-540 (1999; Zbl 0930.19003)]. In fact the authors prove the purity for essentially smooth algebras using the trace formula for the Witt groups of rings and then apply approximation of regular local rings by those algebras. Using the methods, the authors obtain a few other interesting results. For example, they prove injectivity of the natural homomorphism \(W(A_f)\rightarrow W(K)\), where \(f\) is a regular parameter of \(A\).