  This paper consists of four parts. It begins by using the authors's
generalized Schauder formula, [50], and the algebraic multiplicity, $\chi$, of
Esquinas and L\'opez-G\'omez [18,17,40] to package and sharpening all existing
results in local and global bifurcation theory for Fredholm operators through
the recent author's axiomatization of the Fitzpatrick-Pejsachowicz-Rabier
degree, [51]. This facilitates reformulating and refining all existing results
in a compact and unifying way. Then, the local structure of the solution set of
analytic nonlinearities $\mathfrak{F}(\lambda,u)=0$ at a simple degenerate
eigenvalue is ascertained by means of some concepts and devices of Algebraic
Geometry and Galois Theory, which establishes a bisociation between Bifurcation
Theory and Algebraic Geometry. Finally, the unilateral theorems of [40,42], as
well as the refinement of Xi and Wang [63], are substantially generalized. This
paper also analyzes two important examples to illustrate and discuss the
relevance of the abstract theory. The second one studies the regular positive
solutions of a multidimensional quasilinear boundary value problem of mixed
type related to the mean curvature operator.
