Abstract

Polynomial factorization over a field is very useful in algebraic number theory, in extensions of valuations, etc. For valued field extensions, the determination of irreducible polynomials was the focus of interest of many authors. In 1850, Eisenstein gave one of the most popular criterion to decide on irreducibility of a polynomial over Q. A criterion which was generalized in 1906 by Dumas. In 2008, R. Brown gave what is known to be the most general version of Eisenstein-Schönemann irreducibility criterion. Thanks to MacLane theory, key polynomials play a key role to extend absolute values. In this chapter, we give a sufficient condition on any monic plynomial to be a key polynomial of an absolute value, an irreducibly criterion will be given, and for any simple algebraic extension L=Kα, we give a method to describe all absolute values of L extending ∣∣, where K is a discrete rank one valued field.

Highlights

  • Polynomial factorization over a field is very useful in algebraic number theory, for prime ideal factorization

  • The determination of irreducible polynomials was the focus of interest of many authors

  • Brown gave what is known to be the most general version of Eisenstein-Schönemann irreducibility criterion [9]. He showed for a valued field ðK, νÞ and for a monic polynomial f ðxÞ 1⁄4 φnðxÞ þ anÀ1ðxÞφnÀ1ðxÞ þ ... þ a0ðxÞ ∈ Rν1⁄2xŠ, where Rν is a valuation ring of a discrete rank one valuation and φ being a monic polynomial in Rν1⁄2xŠ whose reduction φ is irreducible over ν, aiðxÞ ∈ Rν1⁄2xŠ, degðaiÞ < degðφÞ for every i 1⁄4 0, ... , n À 1, if νðaiÞ ≤ ð1 À i=nÞνða0Þ for every i 1⁄4 0, ... , n À 1 and gcdðνða0Þ, nÞ 1⁄4 1, f ðxÞ is irreducible over the field K

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Summary

Introduction

Polynomial factorization over a field is very useful in algebraic number theory, for prime ideal factorization It is important in extensions of valuations, etc. Brown gave what is known to be the most general version of Eisenstein-Schönemann irreducibility criterion [9]. He showed for a valued field ðK, νÞ and for a monic polynomial f ðxÞ 1⁄4 φnðxÞ þ anÀ1ðxÞφnÀ1ðxÞ þ ... Þ a0ðxÞ ∈ Rν1⁄2xŠ, where Rν is a valuation ring of a discrete rank one valuation and φ being a monic polynomial in Rν1⁄2xŠ whose reduction φ is irreducible over ν, aiðxÞ ∈ Rν1⁄2xŠ, degðaiÞ < degðφÞ for every i 1⁄4 0, ...

Newton polygons
Absolute values
Characteristic elements of an absolute value Let ðK, j jÞ be a non archimedian valued field
Completion and henselization
Key polynomials
Augmented absolute values
Extensions of absolute values
Applications

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