A new proof for the law of decomposition in a general cyclotomic field (Q2545711)

In this paper, the law of decomposition of ideals in a general cyclotomic field is proved without making use of Kummer's theory of decomposition. The proof depends on the following result which is a generalization of Tornheim: Let \(\mathrm{GF}(p^r)\) be a finite field of \(p^r\) elements where \(p\) is a rational prime. Let \(\mathrm{GF}(p^d)\) be a subfield of \(\mathrm{GF}(p^r)\). Then every element of \(\mathrm{GF}(p^r)\) is expressible as a finite linear combination \(\sum l_i\mu_i^m\), \(l_i\in \mathrm{GF}(p^d)\), \(\mu_i\in \mathrm{GF}(p^r)\) if and only if \(m\) has no factor of the form \((p^r-1)/(p^{dd'}-1)\) where \(d'\) is a divisor of \(r/d\) such that \(r/dd'> 1\).