Does Zp Contain Primitive Fourth Roots of Unity?: Investigating p

In summary, the conversation discusses the relationship between prime numbers and primitive roots in the set of integers mod p. It is stated that if p is congruent to 1 mod 3, then Zp contains primitive cube roots of unity. The conversation then considers the opposite statement, asking if p is congruent to 1 mod 4, does Zp contain primitive fourth roots of unity. The solution is that if Zp contains primitive fourth roots of unity, then 4|(p-1), and it is known to be true if q is prime instead of 4. The conversation ends by questioning what values of p would make Zp contain primitive fourth roots of unity.
  • #1
Funky1981
22
0

Homework Statement


p prime, If p=1 ( mod 3) then Zp contains primitive cube roots of unity. Now I am considering which p does Zp contains primitive fourth roots of unity.

opposite way? I mean if p=1(mod4) then Zp contains primitive fourth roots of unity??

2. The attempt at a solution
I can prove that if Zp contains primitive fourth roots of unity, then 4|(p-1) . but how about the opposite way? I mean if p=1(mod4) then Zp contains primitive fourth roots of unity?? I know this statements true if q prime instead of 4. And what values of p does Zp contains primitive fourth roots of unity?
 
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  • #2
Funky1981 said:

Homework Statement


p prime, If p=1 ( mod 3) then Zp contains primitive cube roots of unity. Now I am considering which p does Zp contains primitive fourth roots of unity.

opposite way? I mean if p=1(mod4) then Zp contains primitive fourth roots of unity??

2. The attempt at a solution
I can prove that if Zp contains primitive fourth roots of unity, then 4|(p-1) . but how about the opposite way? I mean if p=1(mod4) then Zp contains primitive fourth roots of unity?? I know this statements true if q prime instead of 4. And what values of p does Zp contains primitive fourth roots of unity?

Hi Funky1981! :smile:

The expression ##p \equiv 1 \pmod 3## means that there is a k such that ##p=3k+1##.

Now suppose g is a primitive root mod p.
Then ##g^{\phi(p)} \equiv g^{3k} \equiv 1 \pmod p##.
Therefore ##g^k## is a cube root of 1 in ##\mathbb Z_p##.Same argument holds for ##p \equiv 1 \pmod 4##...
 

FAQ: Does Zp Contain Primitive Fourth Roots of Unity?: Investigating p

1. What is Zp and what are primitive fourth roots of unity?

Zp refers to the set of integers modulo p, where p is a prime number. Primitive fourth roots of unity, denoted as ω, are complex numbers that when raised to the power of 4, equal 1. These roots are considered "primitive" because they cannot be expressed as a power of a smaller root.

2. Why is it important to investigate if Zp contains primitive fourth roots of unity?

The presence of primitive fourth roots of unity in Zp has significant implications in number theory and algebra. It can help in solving certain equations and understanding the structure of the set of integers modulo p.

3. How can we determine if Zp contains primitive fourth roots of unity?

To determine if Zp contains primitive fourth roots of unity, we need to first calculate the totient function of p, denoted as φ(p). If φ(p) is divisible by 4, then Zp contains primitive fourth roots of unity.

4. Can Zp contain more than one primitive fourth root of unity?

Yes, it is possible for Zp to contain more than one primitive fourth root of unity. For example, if p is a prime number of the form 4k+1, then both ω and -ω will be primitive fourth roots of unity in Zp.

5. How can the presence of primitive fourth roots of unity in Zp be proven?

The presence of primitive fourth roots of unity in Zp can be proven using various mathematical techniques, such as quadratic reciprocity and the Chinese remainder theorem. These techniques involve analyzing the divisibility of φ(p) and the properties of p.

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