Abstract
Let $x^{p^k}-a$ be irreducible over $\mathbf{Q}$. The first part of this paper is to explicitly give the decomposition rules for the factorization of $p$ in the ring of integers of $\mathbf{Q}(a^{1/p^k})$.
As an application of the above we use these results to determine the genus field of $\mathbf{Q}(a^{1/n})$, where $x^n-a$ is irreducible over $\mathbf{Q}$ and we make no restrictions on $a$.
Citation
William Yslas VÉLEZ. "The Factorization of $p$ in $\mathbf{Q}(a^{1/p^k})$ and the Genus Field of $\mathbf{Q}(a^{1/n})$." Tokyo J. Math. 11 (1) 1 - 19, June 1988. https://doi.org/10.3836/tjm/1270134258
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