Alphabeta Math

Number Theory

48 pages in 3 parts

Number theory here starts from the integers as this library constructed them, an ordered commutative ring and nothing more, and asks what their multiplicative structure is. Division with remainder gives the greatest common divisor and Bezout's identity, Bezout gives Euclid's lemma, and Euclid's lemma is the step that makes prime factorisation unique rather than merely possible. Congruence modulo n turns divisibility into an equivalence relation, so the integers modulo n form a ring in which linear equations can be solved; the Chinese remainder theorem splits a composite modulus into its prime powers, Euler's totient counts the units, and the unit group modulo a prime is cyclic, which is what a primitive root is. The collection then asks which residues are squares. The Legendre symbol is multiplicative because that unit group is cyclic, Euler's criterion and Gauss's lemma compute it, quadratic reciprocity relates the symbol for one prime to the symbol for the other, and the Jacobi symbol extends the computation to odd composite moduli.

The number theory track scaffolded to continue this collection rests on exactly these pages, and it splits at the point where they stop: an analytic block that needs the gamma and zeta functions of the complex analysis track, and a local and algebraic block that needs valuation rings, Dedekind domains and ideal classes from commutative algebra. The metric completion of the rationals that produces the p-adic numbers comes from Topology and Real Analysis.

Pathway

The parts run in order. Everything a page needs from this group has been read by the time you reach it, and the level on each row is how many dependency steps into the group that page sits.

  1. Part 1 · Divisibility and primes

    4 pages

    Number theory here starts from the integers as an ordered commutative ring with a division algorithm, so gcds, Bezout's identity and Euclid's lemma lead to unique prime factorisation. The same repeated quotient extraction becomes continued fractions: convergents satisfy determinant identities and sharp approximation bounds, Legendre's criterion characterises best approximants, and eventual periodicity detects quadratic irrationals. For D that periodicity becomes arithmetic in Z[D]: the norm is multiplicative, Pell solutions form a unit group generated by the fundamental solution, parity of the continued-fraction period decides the negative Pell equation, and every nonzero generalized Pell equation breaks into finitely many orbits, making solubility decidable by a finite search.

  2. Part 2 · Congruences and unit groups

    9 pages · after Part 1

    Congruences modulo n, the Chinese remainder theorem, Euler's totient, and primitive roots give quotient-ring arithmetic, while quadratic forms and arithmetic functions organize multiplicative information. For finite K/Q, trace, norm, embeddings, and discriminants lead to the Dedekind property and unique prime-ideal factorization; the different detects ramification and has exponent e−1 in the tame case. Completions supply Qp and Zp. The decomposition page identifies the decomposition group as a prime stabilizer and inertia as the kernel of the residue action; its exact sequence, towers, fixed fields, and Frobenius coset separate ramification, residue degree, and splitting. After good reduction kills inertia, factor degrees give Frobenius cycle lengths.

  3. Part 3 · Quadratic residues and reciprocity

    11 pages · after Part 2

    The Legendre and Jacobi symbols, Euler's criterion, Gauss's lemma, and reciprocity control quadratic residues and the two- and four-square problems. Average orders and Perron inversion lead to Chebyshev and Mertens estimates and the explicit formula, while Hilbert symbols give Hasse--Minkowski and Dirichlet characters give primitive L-function continuation and functional equations. The classical zeta page uses the Hadamard product and a logarithmic-derivative inequality for a zero-free region near 1; a balanced truncated explicit formula yields the stated prime-number-theorem error. Its damped-contour Tauberian route is independent, and the progression result keeps the modulus fixed.