The notion of generic minimum polynomial and generic norm for finite-dimensional strictly power-associative algebras, introduced by \textit{N. Jacobson} [J. Reine Angew. Math. 201, 178--195 (1959; Zbl 0084.03601), and Osaka Math. J. 15, 25--50 (1963; Zbl 0199.07201)] are extended here to infinite-dimensional power-associative algebras which are generically algebraic, i.e., each element \(x\) satisfies a monic polynomial \(m_x(\lambda)=m_i(x)\lambda^i\), where the \(m_i(x)\) are polynomial functions. The generic minimum polynomial is the polynomial of least degree satisfying these properties, and the generic norm is \((-1)^{\text{degree }m_x(\lambda)}\) times times the constant term of \(m_x(\lambda)\). The author proves that any homogeneous polynomial function factors uniquely as a product of a finite number of irreducible polynomial functions, from which the standard properties of the generic norm are carried over. This machinery is then used to settle in the affirmative the generalized Schafer conjecture that every norm on a normed algebra is a product of irreducible factors of the generic norm. This had previously been proved by the author for finite-dimensional algebras [Pac. J. Math. 15, 925--956 (1965; Zbl 0139.25501)], and for certain algebras with forms admitting associative or Jordan composition [J. Algebra 5, 72--83 (1967; Zbl 0153.05901)]. In the latter, the base field had to have enough elements (possibly finite), while in the present paper, it seems to be infinite. Using a generic reduced minimum polynomial \(m_x^0(\lambda)\) [like \(m_x(\lambda)\) except that the result of substituting \(x\) is nilpotent rather than zero], a discriminant \(\delta\) is defined (the discriminant of \(m_x^0(\lambda)\)], and \(A\) is called unramified if \(\delta\neq 0\) identically. This is helpful in carrying over the geometric approach to nonassociative algebras of \textit{H. Braun} and \textit{M. Koecher} [Jordan-Algebren. Berlin etc.: Springer-Verlag (1966; Zbl 0145.26001)] to the infinite-dimensional case. Their homogeneity condition is shown to be equivalent to the condition that the symmetrized algebra \(A^+\) be Jordan. The non-degeneracy condition is given here as the existence of enough associative bilinear forms \(\sigma\) to separate points, i.e., \(\cap \mathrm{Rad}(\sigma)=0\). The algebra is called seminormal if there are enough seminormal forms (symmetric with values in an extension field which annihilate nilpotent elements in any extension) to separate points. Except for certain characteristics, these are flexible noncommutative Jordan algebras, and the simple ones are either finite-dimensional quasi-associative, finite dimensional commutative Jordan or of degree 2 over the center, which is also the description of the simple normed algebras given by the author [1967, loc. cit.].