Nicolette Russell
MATH 311W
Primitive elements modulo prime p
April 28, 2014
Primitive elements in modulo prime p have many important fundamentals. They have
many different theorems and corollaries to help us understand more about them. The theorems
show many different ways to see how the primes are used. They else help you learn more about
primitive elements. Primitive elements are also known as prime roots. In primitive elements
modulo prime p, the p is an odd prime. A prime has exactly two positive divisors which are 1
and p. A way to find prime numbers is to list numbers greater than 2 to any number n. After
making the list, you see that 2 is the first number. Since 2 is the first number, you start crossing
off numbers that are divisible by 2. Once you have crossed out all the numbers divisible by 2,
you look at the first number again. In this case, it would be 3. Now, you start crossing out
numbers divisible by 3. You continue this process until all you have left is numbers that are
divisible with itself or 1. They would be all your prime numbers 2 to n. According a corollary
there are infinitely many prime integers (Humphreys & Prest, 2004, p. 29).
There is one theorem that focuses mostly on primitive elements. Also, there are more
theorems and corollaries that relate to primitive elements. The theorem that focuses mostly
primitive roots is known as Fermat’s Theorem. Fermat’s theorem: let p be a prime and suppose
that a is an integer not divisible by p. Then, ap-1 ≡ 1 mod p (Humphreys & Prest, 2004, p. 63).
Below are some examples of Fermat’s Theorem:
p=3
a=2
p=5
a=2
p=7
a=3
a=2
a=3
a=4
21 ≡ 2 mod 3
21 ≡ 2 mod 5
31 ≡ 3 mod 5
21 ≡ 2 mod 7
31 ≡ 3 mod 7
41 ≡ 4 mod 7
22 ≡ 4 ≡ 1 mod 3
22 ≡ 4 mod 5
32 ≡ 9 ≡ 4 mod 5
22 ≡ 4 mod 7
32 ≡ 9 ≡ 2 mod 7
42 ≡ 16 ≡ 2 mod 7
23 ≡ 8 ≡ 3 mod 5
33 ≡ 12 ≡ 2 mod 5
23 ≡ 8 ≡ 1 mod 7
33 ≡ 6 mod 7
43 ≡ 8 ≡ 1 mod 7
24 ≡ 6 ≡ 1 mod 5
34 ≡ 6 ≡ 1 mod 5
24 ≡ 2 mod 7
34 ≡ 18 ≡ 4 mod 7
44 ≡ 4 mod 7
25 ≡ 4 mod 7
35 ≡ 12 ≡ 5 mod 7
45 ≡ 16 ≡ 2 mod 7
26 ≡ 8 ≡ mod 7
36 ≡ 15 ≡ 1 mod 7
46 ≡ 8 ≡ 1 mod 7
If you have a set called Gp of invertible elements of Zp, the set has p – 1 elements. A
corollary helps us understand more about the elements of Zp. Corollary: let p be a prime and
every non-zero element of Zp is invertible (Humphreys & Prest, 2004, p. 47). The elements in
this would be p – 1. Next, we need to multiply Gp by [a]. Theorem: let n be an integer greater
than or equal to 2. The product of any two elements of Gn is in Gn (Humphreys & Prest, 2004, p.
47). The theorem helps us show [a]Gp is in Gp. No elements that are multiplied by [a] in [a]Gp
with [b] ≠ [c] are equal because if [a][b] = [a][c] then [b] = [c] (Humphreys & Prest, 2004, pp.
63-64).
A corollary helps us see the end result of this proof of Fermat’s theorem. Corollary: let n
be an integer greater than or equal to 2, and let a, b, c be any integers. If n and c are relatively
prime and if ac ≡ bc mod n, then a ≡ b mod n (Humphreys & Prest, 2004, p. 45). Integers are
relatively prime if their greatest common divisor is 1 (Humphreys & Prest, 2004, p. 12). To
prove this, we can write ac ≡ bc mod n as [a]n[c]n = [b]n[c]n. We see that n and c are relatively
prime which means [c]n-1 exists. We get [a]n[c]n[c]n-1 = [b]n[c]n[c]n-1 which is the same as [a]n[1]n
= [b]n[1]n and leads us to [a]n = [b]n. [a]n = [b]n can be rewritten as a ≡ b mod n which gives us
the result we need (Humphreys & Prest, 2004, pp. 45-46).
Primitive elements have a corollary that relates to them. Corollary: let p be a prime
number and let a be any integer not divisible by p. Then the order of a mod p divides p – 1
(Humphreys & Prest, 2004, p. 65). The corollary is proved by a theorem. Theorem: Suppose that
a has order k modulo n then ar ≡ as mod n if, and only if r ≡ s mod k (Humphreys & Prest, 2004,
p. 62). To prove the theorem we start with r ≡ s mod k. r ≡ s mod k has r in the form of s + kt for
an integer t. Now we see that ar = as + kt = as(akt) = as(ak)t ≡ as(1)t mod n. as(1)t mod n is the same
as as mod n (Humphreys & Prest, 2004, p. 62).
The chart above can help us see how this theorem and corollary work. The chart above
shows that 22 ≡ 25 mod 7. As observed on the chart 2 mod 7 has the order of 3. So, we see that 2
≡ 5 mod 3. Now, we observe from 2 mod 7 that p – 1 = 6 and the order is 3 and p divides p – 1 is
3│6 which works. The corollary helps us see that a does not necessarily have an order of p – 1,
but it does have ap-1 ≡ 1 mod p (Humphreys & Prest, 2004, p. 65). The chart above helps us to
see this as well.
Primitive elements have another theorem that relates to them. Theorem: Suppose that p is
a prime and let n be any positive integer then ᴓ(pn) = pn – pn – 1. The theorem is proved by the
integers that are in between 1 and pn that would a factor in common with pn are the integers that
are divisible by p. This is saying that pn = pn – 1p. pn – 1p shows us that there is pn – 1 numbers that
are divisible by p. It also leads us to pn – pn – 1 numbers that are relatively prime to pn
(Humphreys & Prest, 2004, p. 66).
In conclusion, there are many things to learn and know about primitive elements modulo
prime p. The Fermat’s theorem tells us the most about primitive elements. We have other
theorems and corollaries that relate to Fermat’s theorem and primitive elements. They help us
learn easier ways to use them. The theorems and corollaries used are all worked off of each
other. You need them to help you understand everything you need to know about primitive
elements.
Works Cited
Humphreys, J. F., & Prest, M. Y. (2004). Numbers, Groups & Codes. University Press: Cambridge
University Press.