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

MATH437 December 2006

Work in progress: this question page is incomplete, there might be mistakes in the material you are seeing here.

  •  Q1  •  Q2  •  Q3  •  Q4  •  Q5  •  Q6  •  Q7  •  Q8  •  

   Other MATH437 Exams

  •  December 2006  •  December 2011  •  

Question 03
Suppose that $\displaystyle p=8n+5$ is a prime and that a is a quadratic residue modulo p. Show that the two solutions of $\displaystyle x^{2}\equiv a\mod {p}$ are given by one of the following possibilities: $\displaystyle x\equiv \pm a^{n+1}\mod {p}$ or $\displaystyle x\equiv \pm 2^{2n+1}a^{n+1}\mod {p}$.

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 sub-case 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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Euler's Criterion states that $\displaystyle 1=\left({\frac {a}{p}}\right)\equiv a^{(p-1)/2}=a^{(8n+5-1)/2}=a^{4n+2}\mod {p}$ Thus, we must have that $\displaystyle a^{2n+1}\equiv \pm 1\mod {p}$ (since it squares to one and $\displaystyle \pm 1$ are the two unique solutions to $\displaystyle x^{2}\equiv 1\mod {p}$ where uniqueness is due to the fact that p is a prime). Case 1: $\displaystyle a^{2n+1}\equiv 1\mod {p}$. In this case, we have $\displaystyle (\pm a^{n+1})^{2}\equiv a^{2n+2}\equiv a\cdot a^{2n+1}\equiv a\mod {p}$ Case 2: $\displaystyle a^{2n+1}\equiv -1\mod {p}$. In this case, recall that $\displaystyle p\equiv 5\mod {8}$ and so a subcase of quadratic reciprocity says that (along with Euler's criterion again) $\displaystyle -1=\left({\frac {2}{p}}\right)\equiv 2^{(p-1)/2}=2^{4n+2}\mod {p}.$ Using this, we have $\displaystyle (\pm 2^{2n+1}a^{n+1})^{2}\equiv 2^{4n+2}a^{2n+2}\equiv (-1)\cdot a\cdot a^{2n+1}\equiv (-1)a(-1)\equiv a\mod {p}$ and this completes the proof.

Click here for similar questions

MER QGQ flag, MER RH flag, MER RS flag, MER RT flag, MER Tag Euler's criterion, MER Tag Legendre symbol, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag