What question did this study set out to answer?

To classify explicit Galois extensions, Galois group structures, and radical subfield lattices arising from trace–norm polynomials of units in real quadratic fields.

August 19, 2026Open Access

Affine Galois Realizations from Real Quadratic Units: Kummer Defects, Composita, and Radical Lattices

View Full Paper

Ask AI

Bookmark

Authors

DBDavid Betzer

Discussion

Member takes

Implication

Theoretical analysis reveals explicit affine Galois realizations from real quadratic unit radicals, highlighting solvable extensions and radical lattice structures over the rationals.

Key Points

To classify explicit Galois extensions, Galois group structures, and radical subfield lattices arising from trace–norm polynomials of units in real quadratic fields.
Analyzed the splitting fields of the trace–norm polynomial Q_ε,ℓ(x) = (x^ℓ − ε)(x^ℓ − ε̄) using Kummer theory and Dirichlet's unit theorem.
Evaluated Kummer defects, cyclotomic intersections, finite composita fiber-product indices, and Hilbert's irreducibility theorem across prime and composite exponents.
Established that for prime exponents, generic splitting fields yield Galois groups isomorphic to AGL(1,ℓ) × C₂, which reduce to AGL(1,ℓ) under specific quadratic–cyclotomic overlap.
Proved that radical extensions Eₙ = F(ε^(1/n)) satisfy the divisor-lattice identities E_a ∩ E_b = E_gcd(a,b) and E_a E_b = E_lcm(a,b), remaining non-Galois for all exponents n > 2.
Constructed infinite families of pairwise-distinct, solvable Galois extensions of arbitrarily large degree over ℚ with fully determined ramification and discriminant structures.

Cite This Study

David Betzer (2026) studied this question.

synapsesocial.com/papers/6a85638803308d306e2d6bd4 https://doi.org/10.5281/zenodo.21986957

View Full Paper

Ask AI

Bookmark

Implication

Theoretical analysis reveals explicit affine Galois realizations from real quadratic unit radicals, highlighting solvable extensions and radical lattice structures over the rationals.

Affine Galois Realizations from Real Quadratic Units: Kummer Defects, Composita, and Radical Lattices

Discussion

Member takes

Implication

Key Points

Cite This Study

Affine Galois Realizations from Real Quadratic Units: Kummer Defects, Composita, and Radical Lattices

Discussion

Member takes

Implication

Key Points

Cite This Study

Also Consider

Also Consider