[Paper Review] The Chebyshev Exponent

Gene Ward Smith|arXiv (Cornell University)|Sep 13, 2012

History and Theory of Mathematics1 references3 citations

TL;DR

This paper introduces the Chebyshev exponent, a novel algebraic framework analogous to ordinary exponents, using Chebyshev polynomials as the basis. It establishes a formalism for Chebyshev radicals and applies it to construct infinite families of unramified extensions in both real and imaginary quadratic fields, particularly via degree-3 and degree-4 dihedral extensions, proving the existence of infinite families with 3-cycles and 4-cycles in the class group.

ABSTRACT

The analogy between the nth power function and the nth Chebyshev polynomial is pursued, leading to consideration of Chebyshev radicals as analogous to ordinary radicals and Chebyshev exponents to ordinary exponents, and the cosine and hyperbolic cosine as analogs of the exponential function. We then discuss solving polynomial equations in Chebyshev radicals, and apply this to the construction of unramified extensions of quadratic number fields.

Motivation & Objective

• To develop a formal algebraic framework for Chebyshev polynomials analogous to ordinary exponentiation, introducing the 'Chebyshev exponent' notation and its properties.
• To explore the analogy between ordinary powers and Chebyshev polynomials, particularly through the functional composition property and the role of the characteristic equation.
• To apply the Chebyshev exponent formalism to solve polynomial equations in Chebyshev radicals and construct unramified extensions of quadratic number fields.
• To demonstrate the existence of infinite families of unramified extensions in both real and imaginary quadratic fields, specifically for 3-cycles and 4-cycles in the class group.
• To provide explicit polynomial constructions and congruence conditions that ensure unramified behavior over quadratic subfields.

Proposed method

• Define the Chebyshev exponent $ x^{ ext{ extcopyright} n} $ via the recurrence $ x^{ ext{ extcopyright} 0} = 2 $, $ x^{ ext{ extcopyright} 1} = x $, $ x^{ ext{ extcopyright} n} = x x^{ ext{ extcopyright} (n-1)} - x^{ ext{ extcopyright} (n-2)} $, generalizing the power function.
• Establish the fundamental identity $ (z + z^{-1})^{ ext{ extcopyright} n} = z^n + z^{-n} $, which characterizes the Chebyshev exponent and links it to roots of unity and the characteristic polynomial.
• Define the $ ext{ extcopyright} $-operator on polynomials by replacing each monomial $ x^k $ with $ x^{ ext{ extcopyright} k} $, preserving algebraic structure while introducing Chebyshev analogs.
• Use the Chebyshev exponent formalism to analyze the ramification of polynomial extensions, particularly focusing on cubic and quartic dihedral extensions over quadratic fields.
• Derive explicit congruence conditions on parameters $ t $, $ u $, and $ s $ to ensure unramified extensions, based on local behavior at primes dividing the discriminant.
• Apply the resultant method to eliminate $ z $ from $ Q(z) $ and $ z^2 - xz + 1 $, yielding $ (P^{ ext{ extcopyright}})^2 $, which enables algebraic construction of Chebyshev-transformed polynomials.

Experimental results

Research questions

• RQ1 Can a formal algebraic framework be developed for Chebyshev polynomials analogous to ordinary exponentiation, and what are its defining properties?
• RQ2 What are the conditions under which a cubic or quartic polynomial over a quadratic field yields an unramified extension?
• RQ3 How can Chebyshev radicals and exponents be used to systematically construct unramified extensions of quadratic number fields?
• RQ4 What congruence conditions on parameters ensure that a dihedral extension of degree 3 or 4 remains unramified over its quadratic subfield?
• RQ5 Are there infinite families of real and imaginary quadratic fields with nontrivial class group cycles (3-cycles and 4-cycles) that arise from such constructions?

Key findings

• The paper proves the existence of an infinite number of real and imaginary quadratic fields with 3-cycles in the nongenus class group, constructed via cubic polynomials of the form $ x^3 + ux + tu^2 $ under specific congruence conditions on $ u $ and $ t $.
• For $ s = 2 $, the extension $ x^3 + 2ux + tu^2 $ is unramified if $ u $ is divisible by 8, or if $ u $ is even and $ t $ is even, or if $ u $ is odd and $ t ot o 2 mod 4 $.
• For $ s = 3 $, unramified extensions exist when $ u $ is divisible by 3, or when $ u^9 ot o ext{mod } 27 $, or under specific congruence conditions on $ u $ and $ t $ modulo 9 or 27.
• The polynomial $ x^4 - x^3 - tx^2 - x + 1 $ generates an unramified degree-4 dihedral extension over $ extbf{Q}( ext{discriminant}) $ when $ t o -13, -7, 11 mod{30} $, yielding infinite families of such fields.
• The quadratic subfields of the quartic extension are given by $ x^2 - t(t-4) $, $ x^2 - 4t - 9 $, and $ x^2 - t(t-4)(4t+9) $, and ramification is controlled by the divisibility of $ t(t-4)(4t+9) $ by small primes.
• The construction confirms that there exist infinite families of quadratic fields—both real and imaginary—with 4-cycles in the class group, generated by the specified quartic polynomial under congruence conditions on $ t $.