Their topics include divisibility, polynomial congruences, quadratic reciprocity
, the geometry of numbers, and algebraic integers.
It also focuses on broader background, with brief but representative discussions of naive set theory and equivalents of the Axiom of Choice, quadratic reciprocity
, and basic complex analysis.
In section IV the main topic is the law of quadratic reciprocity
One of the most important results of elementary number theory is the so-called law of quadratic reciprocity
, which links prime numbers (those evenly divisible only by themselves and one) and perfect squares (whole numbers multiplied by themselves).
Among specific topics are Fermat, Euler, and quadratic reciprocity
, the Hilbert class field and genus theory, orders in imaginary quadratic fields, modular functions and ring class fields, and elliptic curves.
That content includes divisibility in the natural numbers, linear equations through the ages, the prime numbers, thinking cyclically, Fermat and Euler, cryptography, polynomial congruence, quadratic reciprocity
, Pythagorean triples, sums of squares, Fermat's last theorem, diophantine approximations and Pell equations, primality testing, and mathematical induction.