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Finite local principal ideal rings

Matthé van der Lee

Abstract

Every finite local principal ideal ring is the homomorphic image of a discrete valuation ring of a number field, and is determined by five invariants. We present an action of a group, non-commutative in general, on the set of Eisenstein polynomials, of degree matching the ramification index of the ring, over the coefficient ring. The action is defined by taking resultants.

Finite local principal ideal rings

Abstract

Every finite local principal ideal ring is the homomorphic image of a discrete valuation ring of a number field, and is determined by five invariants. We present an action of a group, non-commutative in general, on the set of Eisenstein polynomials, of degree matching the ramification index of the ring, over the coefficient ring. The action is defined by taking resultants.
Paper Structure (6 sections, 29 theorems)

This paper contains 6 sections, 29 theorems.

Table of Contents

  1. Introduction
  2. Finite local PIRs
  3. Ring structure
  4. The invariant Delta
  5. The unramified case
  6. Further notes

Key Result

Proposition 1

With the notation as above, the following hold.

Theorems & Definitions (58)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • Example 1
  • Lemma 2
  • proof
  • Proposition 3
  • ...and 48 more