What question did this study set out to answer?

To investigate the existence of odd degree polynomials that define hyperelliptic fields with specific properties.

March 14, 2026

On the Existence of Polynomials of Odd Degree Defining a Hyperelliptic Field With Fundamental -Unit of Degree 13 and Periodic Expansion √

Key Points

To investigate the existence of odd degree polynomials that define hyperelliptic fields with specific properties.
Analyzed polynomials of odd degree greater than or equal to 7, excluding degrees 11 and 13.
Utilized Grobner bases for symbolic computations to validate results.
Evaluated properties of fundamental units in hyperelliptic fields.
No odd degree polynomials exist for defining hyperelliptic fields with fundamental units of degree 13, under certain conditions.
All pertinent polynomials of degrees 11 and 13 were identified successfully.
Only finitely many nonequivalent polynomials of degree 5 meet the specified criteria.

Abstract

We proved that there are no polynomials ∈ ℚ, deg is odd, deg ≥ 7, deg ≠ 11,13, for which the corresponding hyperelliptic field ℚ()(√) has a fundamental -unit of degree 13 and for which the expansion of √ into a functional continued fraction is periodic. In the case deg = 11,13, all polynomials with the indicated properties are obtained. It is also proved that there exist at most finitely many pairwise nonequivalent polynomials () ∈ ℚ of degree 5 with such properties. Symbolic computations with Grobner bases play a significant role in proving the main results.

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