Purchase Solution

Proof of two Legendre Symbol identities

Not what you're looking for?

Ask Custom Question

1) Use that (Z/pZ)* is cyclic to directly prove that (-3/p) = 1 when p = 1 (mod 3).

2) If p = 1 (mod 5), directly prove that (5/p) = 1.

I know that with exercise 1, we show that there is an element c in (Z/pZ)* of order 3, and that (2c+1) ^2 = -3.

Similarly for exercise 2, there is a c in (Z/pZ)* of order 5, and [(c + c^4) ^2] + (c + c^4) - 1 = 0.

Purchase this Solution

Solution Summary

In this solution, we provide proof for two algebraic equations.

Solution Preview

Proof:
(1) First, we note that |(Z/pZ)*| = p - 1. From the condition, p = 1 (mod 3), then p - 1 = 3k for some positive integer k. Thus |(Z/pZ)*| = 3k.
Second, since (Z/pZ)* is a cyclic group, then (Z/pZ)* = <x> and x^(3k) = 1. We set c = x^k, then c^3 = x^(3k) = 1 and thus the ...

Purchase this Solution


Free BrainMass Quizzes
Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.