When is the Order Generated by a Cubic, Quartic or Quintic Algebraic Unit Galois Invariant: Three Conjectures

• Published in: Czechoslovak mathematical journal 2020-12, Vol.70 (4), p.905-919
• Main Author: Louboutin, Stéphane R.
• Format: Article
• Language: English
• Subjects: Analysis; Convex and Discrete Geometry; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Ordinary Differential Equations
• ISSN: 0011-4642; 1572-9141
• DOI: 10.21136/CMJ.2020.0019-19

Let ε be an algebraic unit of the degree n ⩾ 3. Assume that the extension ℚ( ε )/ℚ is Galois. We would like to determine when the order ℤ[ε] of ℚ( ε ) is Gal(ℚ( ε )/ℚ)-invariant, i.e. when the n complex conjugates ε 1 , …, ε n of ε are in ℤ[ ε ], which amounts to asking that ℤ[ ε 1 , …, ε n ] = ℤ[ ε ], i.e., that these two orders of ℚ( ε ) have the same discriminant. This problem has been solved only for n = 3 by using an explicit formula for the discriminant of the order ℤ[ ε 1 , ε 2 , ε 3 ]. However, there is no known similar formula for n > 3. In the present paper, we put forward and motivate three conjectures for the solution to this determination for n = 4 (two possible Galois groups) and n = 5 (one possible Galois group). In particular, we conjecture that there are only finitely many cyclic quartic and quintic Galois-invariant orders generated by an algebraic unit. As a consequence of our work, we found a parametrized family of monic quartic polynomials in ℤ[ X ] whose roots ε generate bicyclic biquadratic extensions ℚ( ε )/ℚ for which the order ℤ[ ε ] is Gal(ℚ( ε )/ℚ)-invariant and for which a system of fundamental units of ℤ[ ε ] is known. According to the present work it should be difficult to find other similar families than this one and the family of the simplest cubic fields.