To prepare ourselves for the proof of the Primitive Root Theorem, we will need various results about the order function. One result we will need deals with how ordp(a) behaves.
How Does ord7(a) Behave?
To develop an intuition for how ordp(a) behaves for different values of a, we will choose a specific value for p, and construct a table that lists different values of ord7(a) for different values of a. We will later use this table to identify patterns and develop a lemma based that we will prove.
A Lemma Regarding the Order Function
In this video, we use the table of values that we have constructed for ord7(a) to formulate a lemma that will be useful in understanding primitive roots.
Proof of The Lemma Part I: ab to the power kl is 1
We want to show that ab has order kl. This requires verifying two things, that ab to the power kl gives us one, and that secondly that it is the smallest thing that has that property. In this video, we investigate the easier part: that ab to the power kl is 1.
Proof of The Lemma Part II: kl is the smallest possible exponent
This video provides a complete proof that kl is the smallest possible exponent such that ab to the power kl is congruent to one.
Why We Need The Orders to be Relatively Prime
This video describes what the proof we established gives us, and presents an example that demonstrates why we need (k, l) = 1.[previous] [next]