## Tokyo Journal of Mathematics

### The Factorization of $p$ in $\mathbf{Q}(a^{1/p^k})$ and the Genus Field of $\mathbf{Q}(a^{1/n})$

William Yslas VÉLEZ

#### 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 $a$.

#### Article information

Source
Tokyo J. Math., Volume 11, Number 1 (1988), 1-19.

Dates
First available in Project Euclid: 1 April 2010

https://projecteuclid.org/euclid.tjm/1270134258

Digital Object Identifier
doi:10.3836/tjm/1270134258

Mathematical Reviews number (MathSciNet)
MR947943

Zentralblatt MATH identifier
0664.12003

#### Citation

VÉLEZ, William Yslas. 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 (1988), no. 1, 1--19. doi:10.3836/tjm/1270134258. https://projecteuclid.org/euclid.tjm/1270134258