Let \(K/k\) be a finite Galois extension. Assume that \(k\) is not an algebraic extension of a finite field. Let \(K^*\) be the multiplicative group of \(K\), and let \(\theta(K/k)\) be the product of the multiplicative groups of the proper intermediate fields. The condition that the quotient group \({\mathcal G}=K^*/\theta(K/k)\) be torsion is shown to depend only on the Galois group \(G\). For algebraic number fields and function fields, the authors give a complete classification of those \(G\) for which \({\mathcal G}\) is nontrivial.
Let \(E_ 1,\ldots,E_ t\) be proper intermediate fields of \(K/k\). The authors determine when the quotient group \(K^*/E_ 1^*\ldots E_ t^*\) is either torsion or has a free summand of infinite rank. The authors show that \(K^*=\theta(K/k)\) whenever the Galois group contains \(S_ 4\). They also demonstrate by example that it is possible for \(K^*\) to be the product of the multiplicative groups of three intermediate fields; the situation for \(t=2\) having been previously examined.
The authors also show that \(K^*/\theta(K/k)\) is torsion if and only if the Galois group of \(K/k\) is not a Frobenius complement. If the group \(K^*/\theta(K/k)\) is a nontrivial torsion group, then it is bounded with prime power exponent \(p\) or \(p^ 2\) where \(p\) depends only on the Galois group. When relaxing the requirement that \(K/k\) be a finite Galois extension, the authors show that if \(E_ 1\) and \(E_ 2\) are subfields of \(K\) with \(K/(E_ 1\cap E_ 2)\) algebraic, then \(K^*/E_ 1^*E_ 2^*\) has infinite rank unless \(K\) is either algebraic over a finite field or purely inseparable over \(E_ 1\) or \(E_ 2\). Every field \(K\) of infinite transcendence degree over its prime subfield satisfies \(K^*=E_ 1^*E_ 2^*\) for suitable \(E_ 1\) and \(E_ 2\).