English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

It is a classical problem in algebraic number theory to decide if a numberfield admits power integral bases and further to calculate all generators ofpower integral bases. This problem is especially delicate to consider in aninfinite parametric family of number fields. In the present paper weinvestigate power integral bases in the infinite parametric family of purequartic fields $K=Q(\sqrt[4]{m})$.  We often pointed out close connection of various types of Thue equations withcalculating power integral bases. In this paper we determine power integralbases in pure quartic fields by reducing the corresponding index form equationto quartic binomial Thue equations. We apply recent results on the solutions ofbinomial Thue equations of type $a^4-mb^4=\pm 1\; ({\rm in}\; a,b\in Z)$ aswell as we perform an extensive calculation by a high performance computer todetermine "small" solutions (with $\max(|a|,|b|)<10^{500}$) of binomial Thueequations for $m<10^7$.  Using these results on binomial Thue equations we characterize power integralbases in infinite subfamilies of pure quartic fields. For $1< m< 10^7$ we giveall generators of power integral bases with coefficients $<10^{1000}$ in theintegral basis.

Binomial Thue equations and power integral bases in pure quartic fields