We prove in this article some general steady state bifurcation theorem for a class of nonlinear eigenvalue problems, in the case where algebraic multiplicity of the eigenvalues of the linearized problem is even. These theorems provide an addition to the classical Krasnoselskii and Rabinowitz bifurcation theorems, which require the algebraic multiplicity of the eigenvalues is odd. For this purpose, we prove a spectral theorem for completely continuous fields, which can be considered as a

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... zed version of the classical Jordan matrix theorem and the Fredholm theorem for compact operators. An application to a system of second order elliptic equations is given as well.