Let \(p\) be a prime number and \(L/K\) a normal unramified extension of number fields whose Galois group \(G\) is a \(p\)-group. Let \(E_L\) be the unit group of the ring of integers in \(L\). The authors derive an upper bound for the number \(d_p \widehat{H}^{-1}(G,E_L)\) of generators of the \(-1\)-dimensional Tate cohomology group of the units, which by definition is the quotient of the group \(E_L[N]\) of units killed by the norm \(N = N_{L/K}\) modulo its subgroup \(I_GE_K\), where \(I_G\) is the augmentation ideal.
It is first shown that if \(K\) is a number field with cyclic class group of \(p\)-power order, and if the Hilbert class field \(L\) of \(K\) has class number \(1\), then \(\widehat{H}^{-1}(G,E_L)\) is cyclic. Then the authors prove that if \(K\) is a number field and \(M/K\) an unramified abelian extension such that \(M\) has class number \(1\) and the Galois group of \(M/K\) is a \(p\)-group of rank \(d\), then \(d_p \widehat{H}^{-1}(G,E_L) = \frac16 d(d^2+5)\). Finally in the special case where \(G\) has rank \(2\), explicit generators of \(\widehat{H}^{-1}(G,E_M)\) are given.
For the entire collection see [Zbl 1148.14001].