[Paper Review] Artin--Schreier and Cyclotomic Extensions

Julio Cesar Salas–Torres, Martha Rzedowski–Calderón+1 more|arXiv (Cornell University)|Jun 17, 2013

Algebraic Geometry and Number Theory2 references3 citations

TL;DR

This paper provides a combinatorial proof that any Artin–Schreier extension of a rational congruence function field over a finite field of characteristic p > 0 is contained in the composite of cyclotomic function fields and a constant field extension, without relying on class field theory. The key result explicitly describes the cyclotomic and constant extensions that contain such Artin–Schreier extensions based on the ramification divisor.

ABSTRACT

In this paper we prove that any Artin--Schreier extension of a congruence rational function field is contained in the composite of a cyclotomic function field and a constant field extension that are explicitly prescribed.

Motivation & Objective

• To provide a proof of the containment of Artin–Schreier extensions in cyclotomic and constant field extensions without using class field theory.
• To explicitly describe the cyclotomic and constant extensions that contain a given Artin–Schreier extension of a rational congruence function field.
• To establish a combinatorial characterization of Artin–Schreier extensions with a single ramified prime via degree and divisibility conditions.
• To extend the result to general Artin–Schreier extensions using partial fractions decomposition.
• To verify that the number of such extensions matches the number of extensions inside the specified cyclotomic and constant field composites.

Proposed method

• Use of Artin–Schreier normal form: $ y^p - y = s(T) $, where $ s(T) $ has a prescribed divisor structure.
• Reduction to the case of a single ramified prime divisor via partial fractions decomposition of $ s(T) $.
• Computation of the number of Artin–Schreier extensions with conductor dividing a given power of a prime, using degree and coprimality conditions.
• Establishment of a bijection between such extensions and subextensions of $ K(\Lambda_N)\mathbb{F}_{q^p} $ for appropriate $ N $.
• Application of Lemmas 2.9 and 2.10 to show containment in cyclotomic function fields generated by $ N $-torsion of the Carlitz module.
• Explicit construction of the generator $ y $ as a sum of generators of subextensions, each corresponding to a term in the partial fraction decomposition.

Experimental results

Research questions

• RQ1 Can the containment of Artin–Schreier extensions in cyclotomic and constant field extensions be proven without class field theory?
• RQ2 What is the precise cyclotomic and constant field extension that contains a given Artin–Schreier extension with a single ramified prime?
• RQ3 How does the number of Artin–Schreier extensions with a given ramification divisor compare to the number of subextensions inside a given cyclotomic and constant field composite?
• RQ4 What is the role of the Carlitz module $ \Lambda_N $ in realizing Artin–Schreier extensions?
• RQ5 How does the presence of the infinite prime $ \mathcal{P}_\infty $ affect the cyclotomic conductor in the containment result?

Key findings

• Any Artin–Schreier extension $ F/K $ with $ F = K(y) $, $ y^p - y = s(T) $, is contained in $ K(\Lambda_{\prod_{i=1}^r P_i^{\alpha_i+1}})\mathbb{F}_{q^p} $ when $ \mathcal{P}_\infty $ is unramified.
• When $ \mathcal{P}_\infty $ is ramified, the extension is contained in $ K(\Lambda_{1/T^{\alpha_1+1}})K(\Lambda_{\prod_{i=2}^r P_i^{\alpha_i+1}})\mathbb{F}_{q^p} $.
• The number of Artin–Schreier extensions with conductor dividing $ P^\alpha $ for a prime $ P $ equals the number of such extensions inside $ K(\Lambda_{P^{\alpha+1}})\mathbb{F}_{q^p} $.
• For a rational function field $ K = \mathbb{F}_q(T) $, every Artin–Schreier extension arises as a subextension of a composite of cyclotomic function fields and a constant field extension of degree $ p $.
• The proof relies on partial fractions decomposition to reduce the general case to the single-prime case.
• The result holds even when the infinite prime is ramified, with the cyclotomic conductor adjusted to $ 1/T^{\alpha_1+1} $ in that case.