We continue the investigation of the algebraic and topological structure of the algebra of Colombeau generalized functions with the aim of building up the algebraic basis for the theory of these functions. This was started in a previous work of Aragona and Juriaans, where the algebraic and topological structure of the Colombeau generalized numbers were studied. Here, among other important things, we determine completely the minimal primes ofK and introduce several invariants of the ideals of G(Ω). The main tools we use are the algebraic results obtained by Aragona and Juriaans and the theory of differential calculus on generalized manifolds developed by Aragona and co-workers. The main achievement of the differential calculus is that all classical objects, such as distributions, become C ∞ -functions. Our purpose is to build an independent and intrinsic theory for Colombeau generalized functions and place them in a wider context.
Abstract. This paper shows an elementary and direct proof of the Fundamental Theorem of Algebra, via Bolzano-Weiestrass Theorem on Minima and the Binomial Formula, that avoids: any root extraction other than the one used to define the modulus function over C, trigonometry, differentiation, integration, series, arguments by induction and −δ type arguments. [11], the proof requires a mininum amount of "limit processes lying outside algebra proper". Hence, the proof avoids differentiation, integration, series, angle and the transcendental functions (i.e., non-algebraic functions) cos θ, sin θ and e iθ , θ ∈ R. Another reason to avoid these functions is justified by the fact that the theory of transcendental functions is more profound than that of the FTA (a polynomial result), see Burckel [3]. Also avoided are arguments by induction and − δ type arguments.Many elementary proofs of the FTA, implicitly assuming the modulus function |z| = √ zz, where z ∈ C, assume either the Bolzano-Weierstrass
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