On unramified Galois extensions constructed using Galois representations (Q1849785)

Let \(k\) be an algebraic number field and \(k_{\text{ab}^r}\) the maximal abelian extension of \(k_{\text{ab}^{(r-1)}}\), \(r \in\mathbb{N}\). Suppose that we have an \(n\)-dimensional Galois representation \(\rho\) of \(k\) such that the restriction of \(\rho\) to each decomposition group over a finite set of primes of \(k\) (containing \(\infty\) and the primes at which \(\rho\) ramifies) is triangulable. Let the field \(K\) correspond to the kernel of \(\rho\) and \(2^{r-2} < n \leq 2^{r-1}\). Then it is proved that \(K k_{\text{ab}^r}\) is unramified over \(k_{\text{ab}^r}\). This result is applied to ordinary Galois representations of \(\mathbb{Q}\) to obtain unramified extensions \(K \mathbb{Q}_{\text{ab}^2}\) over \(\mathbb{Q}_{\text{ab}^2}\).