Number Theory

Job: study integers and arithmetic structures — primes, congruences, Diophantine equations, and local–global principles linking \(\mathbb{Z}\) to \(\mathbb{Q}_p\) and \(\mathbb{R}\) — as the base layer under analytic, algebraic, and arithmetic geometry.

Core

Elementary number theory organizes divisibility and modular structure; analytic methods extract prime distribution from \(\zeta\) and \(L\)-functions; algebraic methods replace \(\mathbb{Z}\) by rings of integers \(\mathcal{O}_K\) and class groups. The three layers share the same questions (density of primes, solvability of equations) at increasing structural depth.

Layer Content
Elementary divisibility, primes, Euclidean algorithm, arithmetic functions
Modular \(\mathbb{Z}/n\mathbb{Z}\), CRT, Euler–Fermat, structure of \((\mathbb{Z}/n\mathbb{Z})^\times\)
Analytic \(\zeta(s)\), Dirichlet \(L\), PNT, zero-free regions
Algebraic \(\mathcal{O}_K\), Dedekind domains, class group, class field
Arithmetic geometry schemes over \(\mathbb{Z}\), heights, Galois reps (exit)

Reading map

Page Focus
Foundations Divisibility, primes, Euclidean algorithm, Dirichlet, arithmetic functions
Modular arithmetic \(\mathbb{Z}/n\mathbb{Z}\), CRT, Euler–Fermat, structure of units
Group theory · Rings & fields Algebra language used in modular and algebraic NT
Analytic NT \(\zeta\), \(L\)-functions, PNT
Algebraic NT \(\mathcal{O}_K\), class group, class field
Arithmetic geometry Points, heights, Faltings, étale
Langlands Galois \(\leftrightarrow\) automorphic; modularity
Cryptography Hardness from modular / elliptic arithmetic

Dependency order

  1. Foundations \(\to\) Modular arithmetic.
  2. Parallel: Analytic NT (analysis + Euler products) and/or Algebraic NT (rings + Galois).
  3. Class field \(\to\) Langlands / arithmetic geometry.

Interfaces

Area Contact
Abstract algebra Groups of units; Galois of cyclotomics; rings \(\mathcal{O}_K\)
Complex analysis Contour integration for \(\zeta\); functional equations
Algebraic geometry Spec \(\mathbb{Z}\), curves over finite fields, zeta of varieties
Representation theory Automorphic forms as reps of \(G(\mathbb{A})\)

Notation (paper-standard)

  • \(p,q\) primes; \(n=\prod p_i^{k_i}\); \(\varphi,\mu,\Lambda\) arithmetic functions
  • \(a\equiv b\pmod m\); \((\mathbb{Z}/n\mathbb{Z})^\times\); CRT product decompositions
  • \(\zeta(s)\), \(L(s,\chi)\); \(\pi(x)\); \(\operatorname{ord}_p\), \(\|\cdot\|_p\)
  • \(\mathcal{O}_K\), \(\mathrm{Cl}_K\), \(h_K\), \(\zeta_K(s)\)

Signature results (orientation)

  • Fundamental theorem of arithmetic: unique factorization in \(\mathbb{Z}_{>0}\).
  • Dirichlet: \(\gcd(a,b)=1\Rightarrow\) infinitely many primes \(p\equiv a\pmod b\).
  • CRT: \(\mathbb{Z}/n\mathbb{Z}\cong\prod\mathbb{Z}/p_i^{k_i}\mathbb{Z}\) for \(n=\prod p_i^{k_i}\).
  • Quadratic reciprocity (Gauss): \(\bigl(\frac{p}{q}\bigr)\bigl(\frac{q}{p}\bigr)=(-1)^{(p-1)(q-1)/4}\) for odd primes.
  • Hasse–Minkowski: a quadratic form over \(\mathbb{Q}\) represents \(0\) nontrivially iff it does over \(\mathbb{R}\) and every \(\mathbb{Q}_p\); fails for many higher-degree Diophantine problems (Selmer, etc.).
  • PNT: \(\pi(x)\sim x/\log x\) (equivalent forms via \(\psi(x)\) / zero-free regions of \(\zeta\)).

See it

Local–global for conics. \(x^2+y^2=3\) has solutions in \(\mathbb{R}\) and in every \(\mathbb{Q}_p\)? Checking Hilbert symbols / completion shows failure at \(p=3\) (or an equivalent \(p\)-adic obstruction). The same local–global checklist, upgraded to Galois cohomology \(H^1(\mathrm{Gal},\cdot)\) and Selmer groups, is the modern language for rational points on elliptic curves.

Check

Why does unique factorization fail in \(\mathbb{Z}[\sqrt{-5}]\)?

\(6=2\cdot3=(1+\sqrt{-5})(1-\sqrt{-5})\) with irreducible non-associate factors; the class group / ideal factorization in \(\mathcal{O}_{\mathbb{Q}(\sqrt{-5})}\) restores uniqueness at the ideal level.

State Euler’s criterion and what it computes.

For odd prime \(p\nmid a\): \(a^{(p-1)/2}\equiv\bigl(\frac{a}{p}\bigr)\pmod p\). Computes the Legendre symbol (quadratic residuosity) without listing squares.

Pitfalls

  • “Primes in arithmetic progressions” needs coprimality; \(\gcd(a,b)>1\) yields at most one prime.
  • Hasse principle is special to degree \(2\) (and some other thin classes); counterexamples are the rule for higher genus / higher degree.
  • Analytic continuation of \(\zeta\) is not elementary; formal Euler products alone do not give PNT.

Refs

  • Ireland–Rosen, A Classical Introduction to Modern Number Theory
  • Marcus, Number Fields (algebraic track)
  • Davenport, Multiplicative Number Theory (analytic track)
  • Neukirch, Algebraic Number Theory

Next

Foundations · Modular arithmetic · Analytic NT · Algebraic NT · Cryptography · Langlands