On the units of algebraic number fields (Q1342385)

Let \(K\) be an algebraic number field of degree \(n\) over the rational number field and \(k\) be a proper subfield of \(K\) with \([K: k]=m\). Further, let \(R_ 1\), \(r_ 1\) denote the number of embeddings of \(K\), \(k\) into the real numbers and \(2R_ 2\), \(2r_ 2\) denote the numbers of embeddings of \(K\), \(k\) into the complex numbers. The author studies the difference \(s= (R_ 1+ R_ 2- 1)- (r_ 1+ r_ 2-1)= (R_ 1- r_ 1)+ (R_ 2- r_ 2)\) in the ranks of the nontorsion unit groups of \(K\) and \(k\). In particular, it is proved that \(s\geq r_ 2\). Also, \(s\leq n\) if and only if \(m=2\), \(m=3\) and \(R_ 1\leq r_ 1\), \(m=4\) and \(R_ 1= r_ 2= 0\). The author also specializes the results when \(k\) is generated by a set of roots of unity of \(K\). These results generalize some results of the reviewer [J. Number Theory 7, 385-388 (1975; Zbl 0322.12008)] for the case \(s=0\).