Computational realization of a second cohomology group

Definitions and mathematical statement

Let $K_s$ be a separable closure of the intended field $K$, let $G_K=\operatorname{Gal}(K_s/K)$, and regard $K_s^\times$ as a discrete $G_K$-module with its natural action. The group in the question is continuous Galois cohomology \[ H^2(G_K,K_s^\times). \] For example, a multiplicative continuous two-cocycle $c$ satisfies \[ c(g,h)c(gh,k)=g(c(h,k))c(g,hk), \] and cocycles are identified modulo coboundaries \[ (\delta b)(g,h)=b(g)\,g(b(h))/b(gh). \] The group operation is induced by pointwise multiplication of cocycles.

The supplied problem is to turn this cohomology group into an explicitly computational group. Its usual Brauer-group realization describes classes of finite-dimensional central simple $K$-algebras up to Brauer equivalence, with tensor product as the group law. The research summary specifically invokes number fields and local-invariant/class-field-theoretic descriptions, but the original sentence does not specify the field class or input representation.

A precise computational target would need an effective representation of $K$, finite encodings of cohomology classes, and prescribed operations and tests, such as multiplication, inverse, equality or triviality, and conversion between algebraic, cocycle, and local-invariant data. The source does not require a particular encoding, algorithm, complexity bound, or uniformity in $K$. These possible tasks explain what must be specified; they are not an invented list already solved or a replacement theorem. A theoretical classification through local invariants alone is not silently identified with the requested practical realization.

Short English statement

Develop explicit representations and algorithms for the second Galois cohomology group with multiplicative separable-closure coefficients. The field class and intended operations need specification.

The supplied summary cites the Brauer-group and local-invariant foundations of class field theory, calling them Hasse--Brauer--Noether context, but says no dedicated primary resolution of the practical program was verified. Its OPEN-TRIAGE label describes an unverified research-program request, not an impossibility theorem or a new survey conclusion.

Sources

• Local UnsolvedMath registry, permanent ID 8800007, source code catalogue entry, "Computational realization of a second cohomology group," proposed by Gerhard Frey. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• IPAM, Workshop on Number Theory and Cryptography --- Open Problems (2006), Attachment 6219, PDF page 31, Problem 6, schedule locator https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule; extraction recorded as pdf-and-legacy-ppt visual audit, accessed 2026-07-29.
• The supplied summary names Hasse--Brauer--Noether/class field theory without a separate primary reference or exact computational theorem.
• Source provenance reports inspection of 19 surviving attachments, comprising 17 PDFs and two legacy PPT decks, excluding animation/build repetitions and solved definitions. This is supplied provenance, not an inspection claimed in this draft.
• Current primary-literature status audit and redistribution terms NEEDS_REVIEW for the publicly accessible author/publisher copy, separately from the dataset licence.