ABC implies a Zsigmondy principle for ramification
Abstract
Let $K$ be a number field or a function field of characteristic 0. If $K$ is a number field, assume the $abc$conjecture for $K$. We prove a variant of Zsigmondy's theorem for ramified primes in preimage fields of rational functions in $K(x)$ that are not postcritically finite. For example, suppose $K$ is a number field and $f\in K[x]$ is not postcritically finite, and let $K_n$ be the field generated by the $n$th iterated preimages under $f$ of $\beta\in K$. We show that for all large $n$, there is a prime of $K$ that ramifies in $K_n$ and does not ramify in $K_m$ for any $m<n$.
 Publication:

arXiv eprints
 Pub Date:
 May 2016
 arXiv:
 arXiv:1605.04376
 Bibcode:
 2016arXiv160504376B
 Keywords:

 Mathematics  Number Theory;
 37P05;
 11G50;
 14G25
 EPrint:
 16 pages