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It is very fundamental to study irreducible plane curve singularities inalgebraic geometry. The contents of the paper consist of three parts, calledPart[A], Part[B] and Part[C] with Good Appendix. Our aim is to prove by Part[B]and Part[C] that a complete irreducibility algorithm for the Weierstrasspolynomial of two complex variables and the Puiseux expansions in Part[A] canbe explicitly and rigorously computable in an elementary way, as follows. Forbrevity, Weierstrass polynomials may be written by W-polys throughout thispaper.

Irreducibility algorithm for the Weierstrass polynomials of two complex variables and the Puiseux expansions: Part[A], Part[B], Part[C]