What question did this study set out to answer?

To establish the local and global arithmetic, affine Galois closure structures, and ramification invariants of radical field extensions generated by fundamental units of real quadratic fields.

August 20, 2026Open Access

Affine Galois Closures, Ray-Class Quotients, and Radical-Field Ramification over Real Quadratic Fields

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Authors

DBDavid Betzer

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Implication

Theoretical analysis reveals exact ramification structures and discriminant formulas for radical unit extensions over real quadratic fields, bridging Kummer theory and class field theory.

Key Points

To establish the local and global arithmetic, affine Galois closure structures, and ramification invariants of radical field extensions generated by fundamental units of real quadratic fields.
Constructed splitting fields and trace–norm polynomials for radical extensions of fundamental units over real quadratic fields.
Analyzed local decomposition, inertia groups, lower ramification filtrations, and power-basis indices at primes dividing an odd prime.
Utilized ray-class quotients and Artin characters to classify abelian subfields and evaluate relative and absolute discriminants.
Determined the affine semidirect product Galois group structure and proved that the cyclic subfield embeds into the narrow ray class field modulo the prime and infinite places.
Derived the exact local power-basis index identity linking integral-basis defect directly to wild ramification breaks.
Calculated exact closed-form global discriminant formulas for the resulting parametric polynomial family.

Cite This Study

David Betzer (2026) studied this question.

synapsesocial.com/papers/6a86b56c8a91293e6a1ccb9d https://doi.org/10.5281/zenodo.21987071

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Implication

Theoretical analysis reveals exact ramification structures and discriminant formulas for radical unit extensions over real quadratic fields, bridging Kummer theory and class field theory.

Affine Galois Closures, Ray-Class Quotients, and Radical-Field Ramification over Real Quadratic Fields

Discussion

Member takes

Implication

Key Points

Cite This Study

Affine Galois Closures, Ray-Class Quotients, and Radical-Field Ramification over Real Quadratic Fields

Discussion

Member takes

Implication

Key Points

Cite This Study

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