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Why Eisenstein Proved the Eisenstein Criterion and Why Schönemann Discovered It First

David A. Cox
The American Mathematical Monthly
Vol. 118, No. 1 (January 2011), pp. 3-21
DOI: 10.4169/amer.math.monthly.118.01.003
Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.118.01.003
Page Count: 19
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Why Eisenstein Proved the Eisenstein Criterion and Why Schönemann
Discovered It First
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Abstract

Abstract This article explores the history of the Eisenstein irreducibility criterion and explains how Theodor Schönemann discovered this criterion before Eisenstein. Both were inspired by Gauss’s Disquisitiones Arithmeticae, though they took very different routes to their discoveries. The article will discuss a variety of topics from 19th-century number theory, including Gauss’s lemma, finite fields, the lemniscate, elliptic integrals, abelian groups, the Gaussian integers, and Hensel’s lemma.

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