Science:Math Exam Resources/Courses/MATH437/December 2006/Question 03
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Question 03 

Suppose that is a prime and that a is a quadratic residue modulo p. Show that the two solutions of are given by one of the following possibilities: or . 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

The first fact about a being a quadratic residue should help trigger Legendre symbols. 
Hint 2 

In fact, one might also be reminded of Euler's Criterion. 
Hint 3 

A subcase of quadratic reciprocity associated with 2 will finish the problem off. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Euler's Criterion states that
Thus, we must have that (since it squares to one and are the two unique solutions to where uniqueness is due to the fact that p is a prime). Case 1: . In this case, we have
Case 2: . In this case, recall that and so a subcase of quadratic reciprocity says that (along with Euler's criterion again)
Using this, we have
and this completes the proof. 