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Contents
  1. Hilbert's 12th Problem: Extension of Kronecker-Weber Theorem
  2. Definitions and mathematical statement
  3. Short English statement
  4. Research attempt
  5. Current frontier.
  6. Attack route.
  7. Attempt.
  8. Stress test.
  9. Outcome.
  10. Sources

Hilbert's 12th Problem: Extension of Kronecker-Weber Theorem

Definitions and mathematical statement

A finite extension $L/K$ is abelian if it is Galois and $\operatorname{Gal}(L/K)$ is commutative. Hilbert's twelfth problem seeks an explicit function-theoretic construction of such extensions for every number field $K$, schematically \[ L=K\left(f_1(a_1),\ldots,f_r(a_r)\right), \] where the functions and allowed evaluation points are themselves explicitly specified from arithmetic data attached to $K$. Merely naming elements already known to generate $L$ is not such a construction. The catalogue records a broad construction program, not a uniquely specified function class or one fixed quantified theorem; admissible notions of explicitness must be stated in any proposed resolution.

Formulation and concept references: [S1].

Short English statement

Can all abelian number-field extensions be generated by explicit special values, generalizing roots of unity and complex multiplication?

Research attempt

Current frontier.
This entry records a construction program, not a uniquely fixed allowable function class. I retain its requirement that functions and evaluation points be defined from arithmetic data, not from a generator already known to lie in the desired extension. The ICMS formulation source discusses this program. [S1]. Dasgupta--Kakde prove a substantial totally real result: Brumer--Stark units, together with specified square roots, generate the maximal abelian extension, with a construction involving $p$-adic integration for infinitely many primes. Their theorem is not restricted to imaginary quadratic CM theory, but it also is not a theorem for arbitrary base number fields. [S2], [E1]. Honnor resolves a root-of-unity ambiguity in one of the unit formulas; Dasgupta--Honnor--Spie
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compare three formulas in a 2025 paper. [E2], [E3]. The abstracts and theorem descriptions were checked; no independent reproof of those arithmetic results is claimed.
Attack route.
One possible analytic route generalizes reciprocity laws beyond the signatures covered by existing special-value constructions. Its missing lemma has three parts: algebraicity, the exact class-field action, and separation of distinct Artin classes. A different route enumerates defining polynomials of abelian extensions, but this does not meet the function-theoretic requirement. I pursue the first route far enough to separate its genuinely arithmetic gap from an avoidable primitive-element problem: once finitely many special values separate Artin classes, an explicit bounded recipe produces generators for every intermediate field.
Attempt.
Let $H/K$ be finite abelian with group $G$, $N=|G|$, and suppose, for $r\ge1$, candidate special values $u_1,\ldots,u_r$ have already been proved to belong to $H$. Assume their Galois transforms can be described by a proposed reciprocity law and that \begin{equation}\label{wsref-0-72-33-36-3a-73-65-70-61-72-61-74-69-6f-6e} \bigcap_{i=1}^r\operatorname{Stab}_G(u_i)=\{1\}. \end{equation} These are hypotheses, not consequences of numerical evaluation of the values.

Lemma 36.1 (bounded primitive combination). Some integer $t$ with $0\le t\le(N-1)(r-1)$ makes \[ \alpha_t=\sum_{i=1}^r t^{i-1}u_i \] a generator of $H/K$. Here $t^0=1$, including at $t=0$.

Proof. For each $g\ne1$, the polynomial $P_g(T)=\sum_i(g(u_i)-u_i)T^{i-1}$ in $H[T]$ is nonzero by (r36:separation). It has at most $r-1$ roots in $K$, hence at most that many integer roots. At most $(N-1)(r-1)$ integers are bad for one or more $g$. Among the displayed number of consecutive integers plus one, a good one exists. Its stabilizer is trivial, so its orbit has $N$ elements. Since $H/K$ is Galois, the degree of $\alpha_t$ over $K$ equals its orbit size and $K(\alpha_t)=H$. The case $N=1$ is immediate, and $r=1$ is already covered by the stabilizer condition. $\square$

The polynomial \[ Q_t(X)=\prod_{g\in G}(X-g\alpha_t) \] then lies in $K[X]$ by invariance, has degree $N$, and is irreducible. This supplies an exact algebraic certificate once the reciprocity action and inequalities $g\alpha_t\ne\alpha_t$ are established. It is more informative than an unspecified generic linear combination: the finite range of possible coefficients is given in advance.

Lemma 36.2 (all intermediate fields from the same values). Let $\alpha$ have trivial stabilizer, let $J\le G$, and write \[ P_J(X)=\prod_{j\in J}(X-j\alpha)=X^{|J|}+c_1X^{|J|-1}+\cdots+c_{|J|}. \] Then $K(c_1,\ldots,c_{|J|})=H^J$.

Proof. The coefficients are fixed by $J$. If $g$ fixes all of them, $gP_J=P_J$; since the roots $h\alpha$ are pairwise distinct, this says $gJ=J$, so $g\in J$. Thus the common stabilizer of the coefficients is exactly $J$, and Galois correspondence proves the equality. Because $G$ is abelian, $H^J/K$ is Galois with group $G/J$. Lemma 36.1 can be applied again to the coefficients to give one primitive generator with a bounded integer choice. $\square$

This suggests a modular research program: construct a collection of values for each ray modulus, prove that the reciprocity action separates the ray classes, and let the two lemmas handle primitive generators and subextensions. Finite sums and products of explicit values are again values of explicitly specified finite combinations of the underlying functions, under a convention allowing those arithmetic operations. That convention is stated here; it does not permit inserting the unknown generator as a constant into a newly named function.

An exact toy check is $H={\mathbb{Q}}(\sqrt2,\sqrt3)$ with its four sign-changing automorphisms. The two values $u_1=\sqrt2,u_2=\sqrt3$ separate the group. The choice $t=1$, in the lemma's range $0\le t\le3$, yields \[ \prod_{\epsilon,\delta=\pm1} (X-\epsilon\sqrt2-\delta\sqrt3)=X^4-10X^2+1. \] The companion script checks this polynomial identity and its orbit size exactly. This is a test of the finite mechanism, not a construction for a new class field.

Stress test.
The hard input is (r36:separation), not the primitive-element theorem. An invariant value gives no class-field information: if every $u_i$ lies in a proper fixed field, every proposed combination remains there. Nonzero values of an $L$-function or regulator do not automatically give algebraic numbers with the correct individual Artin transforms. Algebraic independence would also be the wrong objective for numbers that are supposed to lie in a finite extension.

Approximations in ${\mathbb{C}}$ or ${\mathbb{C}}_p$ can sometimes certify that two values differ, but finite precision does not prove algebraicity or exact field membership. A recognition algorithm without a proved degree/height bound is not a proof. Likewise formulas valid only up to unknown roots of unity can alter a stabilizer, so that ambiguity cannot be dropped before applying either lemma. The cited work resolving a specific ambiguity does not resolve every possible one over arbitrary $K$. [E2].

The construction does not identify every number field as totally real or imaginary quadratic, nor assume that passing to a larger field and taking norms preserves enough class information. Norms can send all separating values into a proper subfield; Lemma 36.2 uses the complete orbit polynomial precisely to avoid relying on its constant term alone.

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Outcome.
Outcome: Conditional reduction.

A bounded primitive-generator recipe and an exact intermediate-field extraction lemma have been proved. They clarify a sufficient special-value package, without claiming historical novelty for these elementary algebraic facts. What remains is the central Hilbert-twelfth task for arbitrary $K$: construct appropriate values, prove their algebraicity and reciprocity, and show they separate the required classes. No such universal analytic construction is supplied here, and the broad original program is not declared solved.

Sources