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Algebra · 5 problems

Open problems in Algebra

Every Algebra problem in the Math Lab index — 5 in all. Each carries a precise statement, an honest status (open means open), and a concrete plan for throwing compute or tokens at it.

Birch–Tate conjectureAlgebrauntouched

For a totally real number field F, the order of the tame kernel K₂(O_F) equals |w₂(F) · ζ_F(−1)| — K-theory measured by a zeta value.

Finite lattice representation problemAlgebrauntouched

Is every finite lattice isomorphic to the congruence lattice of a finite algebra?

Green's conjecture on syzygiesAlgebrauntouched

For a smooth canonical curve, the vanishing pattern of the syzygies of its canonical embedding is governed exactly by its Clifford index.

Rota's basis conjectureAlgebrauntouched

Given n disjoint bases of an n-dimensional vector space (or rank-n matroid), the n² vectors can be arranged in an n×n grid whose every row and every column is a basis.

Williamson matricesAlgebrauntouched

For which odd n do Williamson matrices (four symmetric circulant ±1 matrices with A² + B² + C² + D² = 4nI) exist? Each solution yields a Hadamard matrix of order 4n.

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