GA01 — Galois Action on Cyclotomic Fields

Field Value
Domain Galois Theory
System Gal(ℚ(ζ_n)/ℚ) acting on nth roots of unity
Group (ℤ/nℤ)× (multiplicative group of units mod n)
H^k tier H¹
ISA Origami (β→∞)
Status Validated
Opcodes ORBIT · TWIST · LABEL · FLIP
Papers Paper 492, Paper 469

Physical system

The cyclotomic field ℚ(ζ_n) is the field obtained by adjoining a primitive nth root of unity ζ_n = e^{2πi/n} to the rationals ℚ. Its Galois group — the group of field automorphisms that fix ℚ — is:

Gal(ℚ(ζ_n)/ℚ) ≅ (ℤ/nℤ)×

where (ℤ/nℤ)× = {a : 1 ≤ a ≤ n, gcd(a,n) = 1} is the multiplicative group of units modulo n, with |Gal| = φ(n) (Euler totient function).

The action is: σ_a(ζ_n) = ζ_n^a for each a ∈ (ℤ/nℤ)×. This sends each nth root of unity to another nth root of unity — an ORBIT on the set of primitive nth roots. The whole theory of cyclotomic fields is the prototypical example of class field theory, which is the Abelian case of the Langlands programme: every abelian extension of ℚ is a subfield of some ℚ(ζ_n) (Kronecker-Weber theorem).

This entry is the H¹ version of the Langlands programme — no BIND needed for Abelian extensions. LA01 and LA02 add the non-Abelian BIND content.

Target category

GalSet(n) — the category of Gal(ℚ(ζ_n)/ℚ)-sets: finite sets X with an action of (ℤ/nℤ)×. Objects are nth roots of unity; morphisms are (ℤ/nℤ)×- equivariant maps. The Galois ORBIT of a root ζ_n^k (for gcd(k,n)=1) is the full set of primitive nth roots — a single transitive (ℤ/nℤ)× ORBIT.

The minimal polynomial of ζ_n over ℚ is the nth cyclotomic polynomial:

Φ_n(x) = ∏_{gcd(k,n)=1} (x − ζ_n^k)

This is the LABEL eigenvalue of the Galois ORBIT — the polynomial whose roots are precisely one ORBIT.

Interpretation functor

F: C → GalSet(n) defined by:

Opcode F(opcode)
ORBIT Galois action: σ_a : ζ_n^k ↦ ζ_n^{ak mod n}; the group (ℤ/nℤ)× acts by multiplication on the exponent; the orbit of ζ_n (a primitive root) is the set of all φ(n) primitive nth roots of unity
TWIST Kummer extension: √[n]{α} for α ∈ ℚ/(ℚ)^n; generates a degree-n cyclic extension with Galois group ℤ/nℤ; the TWIST adds a geometric phase (nth root extraction) that winds once around the origin in ℂ
LABEL Frobenius element: for a prime p ∤ n, the Frobenius Frob_p ∈ Gal is the unique element satisfying σ_{Frob_p}(ζ_n) = ζ_n^p; equivalently, Frob_p = [p] ∈ (ℤ/nℤ)×; the L-function eigenvalue at p
FLIP Complex conjugation: σ_{-1}(ζ_n) = ζ_n^{-1} = ζ̄_n; the FLIP exchanging ζ_n and its conjugate; generates a ℤ/2ℤ subgroup of Gal

ISA programme 

SETUP:   LABEL[n | choose modulus; n=5 gives 4th-degree extension]
ROOTS:   ORBIT[zeta_n^k for gcd(k,n)=1 | primitive nth roots]
GALOIS:  ORBIT[sigma_a(zeta_n) = zeta_n^a | Galois action, a in (Z/nZ)^*]
FROB:    LABEL[Frob_p = [p] in (Z/nZ)^* | Frobenius at prime p]
POLY:    LABEL[Phi_n(x) = prod(x - zeta_n^k) | cyclotomic polynomial]
KUMMER:  TWIST[adjoin n-th root of alpha | cyclic degree-n extension]
CONJ:    FLIP[sigma_{-1}(zeta_n) = zeta_n^{-1} | complex conjugation]
CLASS:   LABEL[Artin map: p -> Frob_p | prime -> Galois element]
OUTPUT:  LABEL[L(s, chi) = prod_p (1 - chi(p)p^{-s})^{-1} | Dirichlet L-function]

Computable output

  • Cyclotomic polynomial Φ_n(x): the minimal polynomial of ζ_n over ℚ, with degree φ(n). The ORBIT has exactly φ(n) elements — the primitive nth roots. Examples: Φ_5(x) = x⁴+x³+x²+x+1 (degree 4, (ℤ/5ℤ)× ≅ ℤ/4ℤ); Φ_12(x) = x⁴−x²+1 (degree 4, (ℤ/12ℤ)× ≅ (ℤ/2ℤ)²). Computed for all n; coefficients known to be ±1 for n = p, 2p, p² but can be large for n with many factors.

  • Frobenius eigenvalue Frob_p = [p mod n]: for any prime p ∤ n, the Frobenius element is [p] ∈ (ℤ/nℤ)×. Explicitly: ζ_n^{Frob_p} = ζ_n^p. This is the LABEL output of the Galois ORBIT at each prime. The quadratic reciprocity law follows: (p/q) = (q/p)×(−1)^{(p−1)(q−1)/4} is a consequence of the Frobenius element in Gal(ℚ(ζ_{pq})/ℚ).

  • Dirichlet L-function L(s,χ) = Σ_{n≥1} χ(n) n^{−s} = ∏_p (1−χ(p)p^{−s})^{−1}: the LABEL eigenvalue of the Frobenius ORBIT. For a character χ: (ℤ/nℤ)× → ℂ×, the Euler product expresses L(s,χ) as a product over primes — each factor 1/(1−χ(Frob_p)p^{−s}) is the LABEL contribution of prime p to the L-function. L(1,χ) ≠ 0 for χ ≠ 1 (Dirichlet’s theorem: primes in arithmetic progressions).

  • Artin map p ↦ [p]: the explicit isomorphism Gal(ℚ(ζ_n)/ℚ) ≅ (ℤ/nℤ)× is the simplest instance of the Artin reciprocity map — the kernel of the Langlands correspondence for GL_1. The class field theory statement: every abelian extension K/ℚ is determined by its conductor n and a subgroup H ⊂ (ℤ/nℤ)×, with Gal(K/ℚ) ≅ (ℤ/nℤ)×/H. This is the H¹ Langlands correspondence (GL_1 case).

Why H¹ (Abelian Langlands = H¹ Langlands)

The Galois group (ℤ/nℤ)× is Abelian. In the ISA H^k ladder:

  • H⁰ (Origami): fixed points of Galois = ℚ itself; the rational subfield; ORBIT count = 1 (only ℚ is fixed); no Galois action
  • H¹ (Forge): cyclotomic extensions; Abelian Galois group (ℤ/nℤ)×; one- dimensional Galois representations χ: Gal → ℂ×; TWIST = Kummer extensions; class field theory = Abelian Langlands; L-functions = Hecke/Dirichlet L-functions
  • H² (Meld): non-Abelian extensions (GL_2, GL_n Langlands); Galois representations ρ: Gal → GL_n(ℚ̄_ℓ); automorphic forms; L-functions = automorphic L-functions (see LA01, LA02)

Cyclotomic fields are H¹ because their Galois groups are Abelian — the one-dimensional representations χ of (ℤ/nℤ)× are the TWIST content (Dirichlet characters), and the L-function is a LABEL eigenvalue of the TWIST. The Kummer TWIST generates cyclic extensions; complex conjugation is the FLIP.

The passage H¹ → H² requires non-Abelian Galois groups (e.g., S_n for irreducible polynomials of degree n ≥ 5, or GL_2(𝔽_p) for modular forms). The BIND content of H² is the non-commutativity of the Galois action.

Connection to Langlands (LA01/LA02)

GA01 is the foundation for both LA entries:

  • LA01 (Geometric Langlands): generalises from curves over ℚ to curves over 𝔽_q; the cyclotomic character χ_ℓ: Gal → ℤ_ℓ× is the proto-Langlands correspondence for GL_1 over a curve
  • LA02 (Artin/Taniyama-Shimura): the GL_1 Artin map (GA01) lifts to GL_2 (elliptic curves, modular forms); Wiles’s FLT proof = GL_2 Langlands for elliptic curves over ℚ = two-dimensional non-Abelian extension of GA01

Connection to Galois chemistry (Paper 488/492)

GA02 (FeMoco Galois computer) is the molecular realisation of GA01: the G₂ symmetry of FeMoco acts on the 7 iron d-orbital states as a 7-dimensional Galois representation of the FeMoco Galois group. The “Frobenius at prime p” becomes the “spin-orbit coupling at each Fe site” — the orbital counterpart of the number-theoretic Frobenius element.

Validation

  • Gauss (1801), Disquisitiones Arithmeticae: cyclotomic construction of regular 17-gon; Frob_p = [p mod 17] explicitly computed; quadratic reciprocity proved.
  • Kronecker-Weber theorem (Hilbert, 1896): every abelian extension of ℚ is cyclotomic; the complete classification of H¹ Langlands over ℚ.
  • Dirichlet (1837): primes in arithmetic progressions; L(1,χ) ≠ 0 proved; Frobenius density confirmed (Dirichlet density = φ(n)^{-1} per coset).
  • Class field theory (Artin 1927, Chevalley 1940): Artin reciprocity map; abelian Langlands for all number fields; Hecke L-functions = LABEL eigenvalues.

Part of the ISA Zoo. Categorical foundations: Paper 591.