Divisors

Introduction to integer divisibility, modular arithmetic, and their generalizations. Written with the aim to assume as little pre-existing mathematical knowledge as possible of the reader. Structured by the author's sense of 'one thing being motivated by the other,' and as such is neither organized like nor comprehensive enough to be an a reference for "abstract" algebra. Higher algebraic structures are included and developed largely on the basis of how motivated they are by integers, functions of integers, integer-like properties, and sets of equivalence classes of integers. A section entitled "A" is taken to mean "the main theme of this section is A," rather than "this is the full treatment of A."

There are different methodologies to writing proofs. For those who already know math well, the proofs which are often perceived as the "best" are considered so because they use minimal steps for a minimal argument. Such proofs often assume of the reader "the fluency to fill in the gaps," as it were. There are often cases where a mathematician writes something of the form "clearly $A$ implies $B,$" but the purported clarity of the implication is strongly dependent on one's familiarity with the subject.

It is the author's belief that a great volume of the opacity in algebra comes from not wanting to overexplain to the overeducated. One of the most universal phenomena one encounters while learning mathematics, however, is that a concept on first meeting will appear nonsensical and unmotivated until it is given enough exposure and consideration such that it becomes all at once as natural as the sunrise.

Indeed, on the sunrise, governed as it is by such well-understood physical principles, we forget that we once knew so little about it that we thought it was brought to us each day by a charioteer in the sky. Once we know something, the idea of not knowing it becomes alien. This principle motivates the decision to explicate totally elementary steps in many of the proofs included within, since to any given reader, the arrow in $A \implies B$ might as well have been shot by Apollo.

For a proper reference, try Dummit and Foote's Abstract Algebra (3rd edition, 2004), and for a more conventional and less algebraic take on the arc from integer division to quadratic reciprocity, try Burton's Elementary Number Theory (7th edition, 2011).

For other books which proved indispensible in the writing of this one, see also:

• Introduction to Linear Algebra, Serge Lang, 2nd edition, 1986
• Basic Algebra I, Nathan Jacobson, 2nd edition, 1985
• Introduction to Analytic Number Theory, Tom Apostol, 1st edition, 1976