Science:Math Exam Resources/Courses/MATH312/December 2005/Question 09

MATH312 December 2005

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Question 09
Prove In this section, prove the statement given to you. Suppose p is a prime with $p\equiv 1\mod 4$, and r is a primitive root modulo p. Show that -r is also a primitive root modulo 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
Suppose that s is the smallest power of $\displaystyle -r$ such that $\displaystyle (-r)^{s}\equiv 1\mod {p}$. What can we say about s?

Hint 2

Using hint 1, we have that $\displaystyle r^{s}\equiv (-1)^{s}\mod {p}$. As r is a primitive root, we have that $\displaystyle r^{p-1}\equiv 1\mod {p}$ and that $\displaystyle r^{(p-1)/2}\equiv -1\mod {p}$. As these are the only possible values for a power of r to be equivalent to a power of -1, we must have that either $\displaystyle s=p-1$ or $\displaystyle s=(p-1)/2$. Assume the second case and find a contradiction.

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. Assume towards a contradiction that $\displaystyle -r$ is not a primitive root. Then $\displaystyle (-r)^{s}\equiv 1\mod {p}$ for some s smaller than $\displaystyle p-1$. Now, we have that $\displaystyle r^{s}\equiv (-1)^{s}\mod {p}$ (multiply both sides of the previous congruence by $\displaystyle (-1)^{s}$. As r is a primitive root, we have that $\displaystyle r^{p-1}\equiv 1\mod {p}$ and that $\displaystyle r^{(p-1)/2}\equiv -1\mod {p}$. As these are the only possible values for a power of r to be equivalent to a power of -1, we must have that either $\displaystyle s=p-1$ or $\displaystyle s=(p-1)/2$. Suppose towards a contradiction that $\displaystyle s=(p-1)/2$. Then we have that by the above that $\displaystyle -1\equiv r^{(p-1)/2}\equiv (-1)^{(p-1)/2}\equiv 1\mod {p}$ valid since $\displaystyle (p-1)/2$ is even. This is a contradiction and hence $\displaystyle s=p-1$ making $\displaystyle -r$ a primitive root.

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