Cyclotomic fields of class numbers one and two

Acreman, Dennis (1978) Cyclotomic fields of class numbers one and two. Masters thesis, Memorial University of Newfoundland.

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Abstract

We find all fields of type Q(exp 2πi/m) with class number hm equal to one or two. We derive various class number formulas and properties associated with these formulas and use these in determining class numbers of cyclotomic fields. The integer hm decomposes as the product h*m h+m of two integers where h+m is the class number of Q(cos 2π/m). We find when h*m = 1 and show that for such m, h+m = 1 also. There are 29 distinct full cyclotomic extensions of Q with class number one and m = 90 is the largest integer for which hm = 1. We also find when h*m = 2 and show that for such m, h+m = 1. There are 2 distinct full cyclotomic extensions of Q with class number two; namely, m = 39 and 56.

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