A remark on the number of Frobenius classes generating the Galois group of the maximal unramified extension

초록

Assume that $K$ is a number field and $K_{ur}$ is the maximal unramified extension of it. When $\text{Gal}(K_{ur}/K)$ is an infinite group. It is known that $\text{Gal}(K_{ur}/K)$ is generated by finitely many Frobenius classes of $\text{Gal}(K_{ur}/K)$ by Y. Ihara. In this paper, we will give the explicit number of Frobenius classes which generate whole group $\text{Gal}(K_{ur}/K)$.

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