On the quartic residue symbol of totally positive quadratic units
type: Item
instance of: scholarly article
title: On the quartic residue symbol of totally positive quadratic units
zbMATH Open document ID: 0604.12003
author: Noburo Ishii
published in: Tokyo Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let m be squarefree positive integer and \(\epsilon_ m\) \((>1)\) the fundamental unit of the real quadratic field \({\mathbb{Q}}(\sqrt{m})\). For a rational prime p satisfying \[ (\frac{- 1}{p})=(\frac{m}{p})=(\frac{\epsilon_ m}{p})=1, \] the biquadratic symbol \((\frac{\epsilon_ m}{p})_ 4\) is defined. The author evaluates this symbol in the following cases: \[ m=qq': q\equiv 3,5\quad (\bmod 8),\;q'\equiv 3\;(\bmod 4),\;(\frac{q}{q'})=-1,\;m=2q: q\equiv 3\;(\bmod 8),\;m=q: q\equiv 3,7,11\;(\bmod 16), \] where q and q' denote prime numbers.
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11A15
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 3975038
zbMATH Keywords: quadratic residue symbols
zbMATH Keywords: fundamental unit
zbMATH Keywords: real quadratic field
zbMATH Keywords: biquadratic symbol
reviewed by: Kenneth S. Williams
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270150975
OpenAlex ID: W2025774645
DOI: 10.3836/TJM/1270150975
description: scientific article; zbMATH DE number 3975038