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Roots of polynomials over division rings

Alina G. Goutor, Sergey V. Tikhonov

Abstract

In this paper, we study properties of polynomials over division rings. Moreover, we present formulas for finding roots of some polynomials

Roots of polynomials over division rings

Abstract

In this paper, we study properties of polynomials over division rings. Moreover, we present formulas for finding roots of some polynomials
Paper Structure (3 sections, 9 theorems, 32 equations)

This paper contains 3 sections, 9 theorems, 32 equations.

Table of Contents

  1. Introduction and preliminary results
  2. Proof of Theorem
  3. Roots of polynomials

Key Result

Theorem 1

(FaMiSeSo17). Let $P(x) = (x-q_n)\dots(x-q_1)$, where $q_1,\dots,q_n \in \mathbb{H}$. If the conjugacy classes $[q_k]$ are distinct, then the polynomial $P(x)$ has exactly $n$ roots $\zeta_k$ which are related to the elements $q_k$ as follows: and $\overline{P}_k(x)$ is the conjugate polynomial of $P_k(x)$.

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 6
  • Remark 7
  • Lemma 8
  • Theorem 9
  • Corollary 10
  • ...and 2 more