It's still unknown whether there are [infinitely many](https://en.wikipedia.org/wiki/List_of_number_fields_with_class_number_one) number fields of class number 1. So I decided to investigate more number fields uncovered in the Wikipedia article. $${}$$ - [Littlewood Polynomials](https://en.wikipedia.org/wiki/Littlewood_polynomial) , i.e. $$\mathbb{Q}[x]/f(x)$$ where $$f(x)$$ is a Littlewood polynomial. Conjecture: There exists $$δ>0$$, such that for sufficiently large degree, the probability for a [randomly-chosen](https://en.wikipedia.org/wiki/Probabilistic_method) Littlewood polynomial to generate a class-1 field $$\mathbb{Q}[x]/f(x)$$ is larger than $$δ$$. The way to randomly choose Littlewood polynomials is up to interpretation: it's not sure how one could avoid reducible polynomials. Computational results: | Degree | Number of fields$$^*$$ with class number > 1 | | ---- | ---- | | ≤9 | 0 | | 10 | 2 | |11|0| |12|4| |13| 8| |14 |45| |15 |46| |16 |135| |17 |212| |18 |452| |19 |766| |20 |1122| *Different Littlewood polynomials could generate the same field, e.g. $$x^2-x-1$$ and $$x^2+x-1$$, so there's a difference between counting the number of polynomials and the number of fields. $$\\$$ $$\\$$ - $$\mathbb{Q}[x]/(x^n+x+1)$$. (Requires x^n^+x+1 to be reducible, i.e. x≠2 mod 3 unless x=2) All computed cases have class number 1. The $$n$$s in the cases are up to 70 except 69. Remark: To compute the class number, one can [use](https://en.wikipedia.org/wiki/Class_number_formula) the Dedekind L function(hard, since the field is generally non-abelian) and the regulator. An empirical formula for the regulator: $$\log(\text{regulator})=A+Bn+Cn\log{n}$$. Computational results: |n|REGULATOR|Class number |----|----| 2|1 |1 3 |0.382245085840036|1 4 |0.337377803571562|1 6 |0.658529208858349|1 7 |1.28891714323854|1 9 |4.58756375116217|1 10 |7.34813790494827|1 12 |43.0124997055349|1 13 |117.917198365552|1 15 |548.874139448263|1 16 |1339.39466430348|1 18|9114.45867514349|1 19|40842.9289743928|1 21|312816.6014877311|1 22|685583.0462776966|1 24|9163307.175032053|1 25|36955124.38960424|1 27|314436524.8339556 |1 28|832511028.3094091 |1 30|11086740597.223595 |1 31|60402933903.83712 |1 33|794597544869.8735 |1 34|1946991838336.107 |1 36|42496810214502.4 |1 37|232694775426930.7|1 39|1954406735071007.5 |1 40|9483736502678452.0|1 42|1.22671527299859e17|1 43|8.65244826944888e17|1 45|1.81327874783812e19|1 46|5.12503081683112e19|1 48|1.44282942692530e21|1 49|8.96333031274895e21|1 51|1.39007666627654e23|1 52|5.53110242701541e23|1 54|1.37500354842092e25|1 55|5.98268106963661e25|1 57|2.10334315815568e27|1 58|7.03923877738823e27|1 60|2.40946814771980e29|1 61|1.45431932648604e30|1 63|2.90399143790153e31|1 64|1.36851987479281e32|1 66|3.34395953400078e33|1 67|2.38125833961797e34|1 69|?|? 70|2.97700584009327e36|1