Local class field theory is treated by means of cohomology theory. Let \(L/K\) be a Galois extension with Galois group \(\mathfrak L\). Let \(\mathfrak H\) be an invariant subgroup of \(\mathfrak L\), and \(F\) be the corresponding subfield of \(L\). The lifting \(\lambda\) of the Galois 2-cohomology group \(H^2(\mathfrak L/\mathfrak H, F^*)\) \((F^*\) being the multiplicative group of \(F)\) into \(H^2(\mathfrak L, L^*)\) is an (into-) isomorphism, and the kernel of the restriction \(\rho\) of \(H^2(\mathfrak L, L^*)\) into \(H^2(\mathfrak H, L^*)\) coincides with \(\lambda (H^2(\mathfrak L/\mathfrak H, F^*))\). These are known from the theory of crossed products and a purely cohomological proof to the former was given by \textit{S. Eilenberg} and \textit{S. MacLane} [Ann. Math. (2) 48, 326--341 (1947; Zbl 0029.34101), but the author proves the latter half also cohomologically (cf. a later paper \textit{J.-P. Serre}, C. R. Acad. Sci., Paris 231, 643--646 (1950; Zbl 0038.36501)], too). Then, for two Galois extensions \(\mathfrak F_1, \mathfrak F_2\) of \(K\), the above two facts are used to transfer a Galois 2-cohomology class in \(F_1/K\) split by \(F_2\) to a such in \(F_2/K\), and this procedure is used to obtain a ``principal'' 2-cohomology class for a Galois extension \(F\) of a \(\mathfrak p\)-adic number field \(K\), starting from such a one for an unramified, cyclic extension of \(K\). Then a ``principal'' homomorphism of the Galois group \(\mathfrak F\) of \(F/K\) into the norm-class group \(K^*/N_{F/K}(F^*)\) is defined by means of the principal cohomology class and a construction introduced by the reviewer [Math. Ann. 112, 85--91 (1935; Zbl 0012.39004)]. A theorem of \textit{E. Witt} [J. Reine Angew. Math. 173, 191--192 (1935; Zbl 0012.14805)] and \textit{Y. Akizuki} [Math. Ann. 112, 566--571 (1936; Zbl 0013.29302)] is used to see the relationship of the principal homomorphisms for \(L/K\) and \(F/K\) with \(L\supset F\). Then the main theorems of local class field theory, including the existence theorem, are established more or less similarly as in [\textit{C. Chevalley}, J. Reine Angew. Math. 169, 140--157 (1933; Zbl 0006.29202)]. The reviewer's note: The purely cohomological method in the present paper has served to pave the road for later cohomological studies of global class field theory; cf. \textit{T. Nakayama} [Ann. Math. (2) 55, 73--84 (1952; Zbl 0046.03802)], Hochschild-Nakayama [Ann. Math. (2) 55, 348--366 (1952; Zbl 0047.03801)], \textit{J. Tate} [Ann. Math. (2) 56, 294--297 (1952; Zbl 0047.03703)].