It’s still unknown whether there are infinitely many number fields of class number 1.

So I decided to investigate more number fields uncovered in the Wikipedia article.

• Littlewood Polynomials , i.e. where is a Littlewood polynomial.

Conjecture: There exists , such that for sufficiently large degree, the probability for a randomly-chosen Littlewood polynomial to generate a class-1 field is larger than .

The way to randomly choose Littlewood polynomials is up to interpretation: it’s not sure how one could avoid reducible polynomials.

Computational results:

*Different Littlewood polynomials could generate the same field, e.g. and , so there’s a difference between counting the number of polynomials and the number of fields.

• .

(Requires xⁿ+x+1 to be reducible, i.e. x≠2 mod 3 unless x=2)

All computed cases have class number 1. The s in the cases are up to 70 except 69.

Remark: To compute the class number, one can use the Dedekind L function(hard, since the field is generally non-abelian) and the regulator.

An empirical formula for the regulator: .

Computational results: