Introduction to Arithmetic Factorization and Congruences from the Standpoint of Abstract Algebra

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Introduction .1 I. Associative algebraic systems. 5 1.1. Some references to associative algebra. 5 1.2. Some definitions. 5 1.3. Three theorems on semigroups. 6 II. Semigroups with a unique basis, and primes.. 8 2.1. General uniquely factorable semigroups . 8 2.2. Unique factorization in the multiplicative semigroup of natural numbers. 8 III. Ideals and congruences in the ring of rational integers . .12 3.1. Principal ideals involving the rational integers . . . 12 3.2. Elements of arithmetic congruence theory . . . 12 3.3. An algorithm for a solution of a linear congruence 14 IV. The ring of residue classes modulo m. . .17 4.1. Residue classes ..17 4.2. An application of semigroups to a linear congruence 17 4.3. Absolutely distinct solutions of polynomial equations ill com- mutative rings .17 4.4. A theorem on the product of the distinct elements of a finite Abelian group .18 V. The additive group of the ring of residue classes modulo i . . . 20 5.1. On the generators of cyclic groups with application to the addi- tive group modulo m .20 5.2. Simple properties of the totieit .22 VI. The multiplicative group @(m) of residue classes Ca modulo m; (a, m) = 1 24 6.1. The theorems of Euler and Wilson . .24 6.2. Criterion for the solution of a quadratic congruence modulo p . 25 6.3. The Minkowski-Thue theorem on linear congruences . 25 6.4. The expression of p=4n+1 as the sum of two squares 26 6.5. Repetitive sets with application to Euler's theorem . 26 6.6. A basis theorem for finite Abelian groups . .27 6.7. A criterion for cyclic groups ..29 6.8. The number of solutions of ax=b (mod m), (a, m) d, with application to cyclic groups ..30 6.9. Primitive roots modulo m ..31 6.10. A theorem on the cyclic subgroup of @(m) of maximal order, with application to Carmichael numbers . .34 6.11. Integral homogeneous symmetric functions defined over repeti- tive sets in commutative rings with units . .36 6.12. Direct product of semigroups with application to @(m) 37 VII. Quadratic reciprocity .42 7.1. Simple properties of the Legendre symbol .42 7.2. Gauss' lemma .43 7.3. The law of quadratic reciprocity .44 VIII. The semigroup of the nonunits of the multiplicative semigroup of resi- due classes modulo m ..48 8.1. On the groups contained in finite cyclic semigroups 48 8.2. The cyclic semigroups generated by nonunit residue classes . 48 8.3. Unique factorization modulo m . .49 IX. The semiring formed by certain generalized residue classes . . 52 9.1. Some finite commutative semirings . .52 9.2. The generalized residue classes