Number Theory
Class Field Theory
Цели урока
- Understand the main theorem: Gal(K^ab/K) is isomorphic to the idele class group via the Artin map
- Know the Artin symbol and its relation to Frobenius elements
- Understand the Hilbert class field and splitting of primes
- See how quadratic reciprocity is a consequence of Artin reciprocity
Предварительные знания
- L-functions and BSD
- Galois theory
- Basic algebraic number theory
Which primes are sums of two squares? Fermat answered this: primes p = 2 or p congruent to 1 mod 4. Class field theory explains why - and generalizes this to arbitrary number fields.
- Cryptography: abelian extension structure determines group orders for elliptic curves over number fields
- Number sieve algorithms: factorization via algebraic number fields uses splitting of primes
- AG codes: Goppa codes exploit prime splitting in function fields for error correction
- Post-quantum crypto: lattice-based systems use ring of integers in CM fields characterized by class field theory
From Gauss to Artin: 120 years toward a unified reciprocity law
Gauss proved quadratic reciprocity in 1801 - one of the most beautiful results in number theory - and gave 8 proofs. He sensed a deeper structure but lacked the tools. Hilbert in 1900 posed as his 12th problem: explicitly describe all abelian extensions of a number field. Takagi (1920) and Artin (1927) completed the theory. Chevalley (1940) gave the modern formulation using ideles. The Langlands program - still open - is the non-abelian generalization.
Artin Reciprocity and Abelian Extensions
In 1927, Emil Artin proved his reciprocity law, unifying all previously known reciprocity laws (Gauss, Cubic, Biquadratic) into a single formula. Modern elliptic-curve cryptosystems protecting 5 trillion USD of daily transactions rely on the structure of abelian extensions of number fields.
The Legendre symbol (p/q) is a special case of the Artin symbol for the extension Q(sqrt(q))/Q. All classical reciprocity laws - quadratic, cubic, biquadratic - are consequences of Artin reciprocity. One theorem subsumes 150 years of number theory.
What does the main theorem of class field theory say about Gal(K^ab/K) for a number field K?
Local Class Field Theory
Local class field theory describes abelian extensions of local fields (Q_p and its finite extensions). It is simpler than the global theory - no Hasse principle issues - and serves as the building block for the global theory via the idele construction.
Kronecker-Weber theorem - the global analog: every abelian extension of Q is contained in a cyclotomic field Q(zeta_n). Proved via local class field theory applied at all primes simultaneously.
Splitting in Q(sqrt(p)) via local theory
Quadratic reciprocity from class field theory
An odd prime q splits in Q(sqrt(p)) if and only if p is a square in Q_q^x, equivalently if the Legendre symbol (p/q) = 1. By Artin reciprocity applied to the product formula over all places, this gives the classical law of quadratic reciprocity: (p/q)(q/p) = (-1)^{(p-1)(q-1)/4}. The entire classical theory is subsumed in one application of the product formula.
What is Gal(Q_p^ab / Q_p) isomorphic to by local class field theory?
Local class field theory: Gal(Q_p^ab/Q_p) is isomorphic to Q_p^x via the local Artin map. The unramified part (generated by Frobenius) corresponds to Z, the inertia corresponds to Z_p^x.
Ideles and the Global Theory
Global class field theory unifies all local information simultaneously using ideles - elements of the restricted direct product of local groups. This is the correct language for the global Artin map and the Galois correspondence for all abelian extensions of a number field.
| Object | Local version | Global version |
|---|---|---|
| Field | Q_p (local) | K (number field) |
| Group | K_v^x | A_K^x / K^x (idele classes) |
| Extensions | Q_p^ab | K^ab |
| Artin isomorphism | K_v^x -> Gal(K_v^ab/K_v) | A_K^x/K^x -> Gal(K^ab/K) |
The Artin reciprocity condition (product of all local symbols equals 1) is nontrivial: it is equivalent to the diagonal embedding K^x lying in the kernel of Art_K. This is the formalization of the idea that global fields are controlled by local data everywhere at once.
Why are ideles needed instead of ordinary fractional ideals in global class field theory?
Ideles are the right language because they simultaneously incorporate all places, including archimedean. The global Artin map A_K^x/K^x -> Gal(K^ab/K) is the product of local maps, and the Artin reciprocity follows from the product formula K^x maps to 1.
Hilbert Class Field and Splitting of Primes
The Hilbert class field H_K is the maximal unramified abelian extension of K. Its Galois group is isomorphic to the ideal class group Cl(K) via the Artin map. This gives a precise answer to the question: which primes split completely in H_K?
Q(sqrt(-5)): non-factorial ring and splitting
Class number 2 example
The ring Z[sqrt(-5)] is not a unique factorization domain: 6 = 2*3 = (1+sqrt(-5))*(1-sqrt(-5)) are two distinct factorizations. The class group Cl(Q(sqrt(-5))) has order 2. The Hilbert class field is Q(sqrt(-5), sqrt(-1)) = Q(sqrt(-5), i). A rational prime p splits completely in this field if and only if the ideal (p) in Z[sqrt(-5)] is principal - that is, p has no obstruction to unique factorization.
A prime p splits completely in the Hilbert class field H_K if and only if:
p splits completely in H_K iff the Artin symbol Frob_p = [(p)] in Cl(K) is trivial, iff (p) is a principal ideal. This directly connects prime splitting to the factorization obstruction measured by Cl(K).
Connections to other topics
Class field theory is the central result of algebraic number theory, connecting analysis, algebra, and geometry.
- Galois theory — Related topic
- Algebraic K-theory — Related topic
- Etale cohomology — Related topic
Итоги
- Main theorem: Art_K: A_K^x / K^x -> Gal(K^ab/K) is a topological isomorphism
- Artin symbol (L/K, p) = Frob_p for unramified prime ideals p
- Hilbert class field H_K: Gal(H_K/K) = Cl(K); p splits completely iff (p) is principal
- Quadratic reciprocity follows from Artin reciprocity applied to Q(sqrt(p))/Q
- Kronecker-Weber: every abelian extension of Q is contained in a cyclotomic field
Вопросы для размышления
- Why does class field theory break down for non-abelian extensions, and what new structure is needed for the Langlands program?
- How does the Hilbert class field encode the obstruction to unique factorization in the ring of integers O_K?
- What is the precise connection between the local Artin map at a single prime and the global Artin map built from all primes simultaneously?
Связанные уроки
- nt-23 — L-functions and BSD motivate the structure of abelian extensions
- nt-25-iwasawa-theory — Iwasawa theory studies class groups along Zp-extension towers
- nt-26-langlands — The Langlands program is the non-abelian generalization of class field theory