Question

Let p be a prime of the form 4^n + 1. Show that 3 is not a quadratic residue modulo p. Please use the Euler's Criterion and Quadratic Reciprocity Theorem accordingly.

Name: let p be a prime of the form 4n 1 show that 3 is not a quadratic residue modulo p please use the eulers criterion and quadratic reciprocity theorem accordingly 97994
Uploaded: 2023-09-05T14:33:30-08:00
Duration: 3 min 22 s
Channel: Madhur L
Description: let p be a prime of the form 4n 1 show that 3 is not a quadratic residue modulo p please use the eulers criterion and quadratic reciprocity theorem accordingly 97994

          Let p be a prime of the form 4^n + 1. Show that 3 is not a
quadratic residue modulo p.
Please use the Euler's Criterion and Quadratic Reciprocity
Theorem accordingly.

Added by William N.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

09/05/2023

Step 1

This means that $p \equiv 1 \pmod{4}$. Now, let's use Euler's Criterion, which states that for any integer $a$ and an odd prime $p$, we have: $$a^{\frac{p-1}{2}} \equiv \left(\frac{a}{p}\right) \pmod{p}$$ where $\left(\frac{a}{p}\right)$ is the Legendre Show more…

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Let p be a prime of the form 4^n + 1. Show that 3 is not a quadratic residue modulo p. Please use the Euler's Criterion and Quadratic Reciprocity Theorem accordingly.