Pollack introduces algebraic number theory to readers who are familiar with linear algebra, commutative ring theory, Galois theory, a little abelian groups theory, and elementary number theory up to and including the law of quadratic reciprocity
Their topics include divisibility, polynomial congruences, quadratic reciprocity
, the geometry of numbers, and algebraic integers.
Finally, we define the notion of quadratic reciprocity
equivalence and prove Theorem 1.
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).
There he completed most of his first major work, Disquisitiones Arithmeticae, including its two proofs of the law of quadratic reciprocity (his "golden theorem") in 1796.
The remaining four of his six different proofs of the law of quadratic reciprocity are believed to have been found by him by 1808.
In fulfillment of this standard, he provided the first correct proofs of such landmark results as the fundamental theorem of algebra (a theorem in complex analysis that states essentially that every polynomial equation has a complex root), and the law of quadratic reciprocity, which is concerned with the solvability of certain pairs of quadratic congruences and is crucial to the development of number theory.
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.
Some third of the material is concerned with biographical and other contextual issues, while the bulk of the selections focus on particular aspects of Euler's contributions to mathematics, including infinite series, the zeta functions, Euler's constant, differentials, multiple integrals, the calculus of variations, the pentagonal number theorem, quadratic reciprocity
, and the fundamental theorem of algebra.