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When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures

Stéphane R. Louboutin (2020)

Czechoslovak Mathematical Journal

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...

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