The quadratic character of 1+\sqrt{2} and an elliptic curve

TL;DR

This paper presents a new criterion for the quadratic character of 1+√2 when p ≡ 1 mod 8, using elliptic curves, offering an alternative to traditional algebraic number theory methods.

Contribution

It introduces a novel elliptic curve-based criterion for the quadratic character of 1+√2, contrasting with previous algebraic number theory approaches.

Findings

Established a criterion linking quadratic character to elliptic curves.

Provided an alternative proof method using elliptic curves.

Connected the quadratic character to the class number of rdp;

Abstract

When p is congruent to 1 mod 8, we have a criterion of the quadratic character of 1+\sqrt{2}, which is related to the class number of \Q(\sqrt{-p}). In this paper, we obtain a similar criterion using an elliptic curve, which contrasts to the proof using algebraic number theory for the old one.