A Note on Cyclotomic Fields.
1. Introduction. The recent article [1] gives explicit evaluations for exponential sums of the form where χ is a non-trivial additive character of the finite field , odd, and . In my dissertation [5], in particular in [4], I considered more generally the sums S(a,N) for all factors N of . The aim of the present note is to evaluate S(a,N) in a short way, following [4]. We note that our result is also valid for even q, and the technique used in our proof can also be used to evaluate certain...
We show that any factorization of any composite Fermat number into two nontrivial factors can be expressed in the form for some odd and , and integer . We prove that the greatest common divisor of and is 1, , and either or , i.e., for an integer . Factorizations of into more than two factors are investigated as well. In particular, we prove that if then and .
For an algebraic number field and a prime , define the number to be the maximal number such that there exists a Galois extension of whose Galois group is a free pro--group of rank . The Leopoldt conjecture implies , ( denotes the number of complex places of ). Some examples of and with have been known so far. In this note, the invariant is studied, and among other things some examples with are given.