# Is α a Primitive Element Modulo p Given α^q ≡ -1 mod p?

• hope2009
In summary, the conversation discusses a scenario where p and q are odd primes and p=2q+1. It is stated that α∈ Z_p^*,α≢±1 mod p and the goal is to prove that α is a primitive element modulo p if and only if α^q≡-1 mod q. The conversation also brings up the concept of "Sofie Germain Prime" and Fermat's Little Theorem. It is mentioned that if a number is a primitive root, then all its powers Modulo p must differ, giving us the p-1 residue system. Finally, there is an error pointed out regarding the equation a^q \equiv -1 \bmod p.
hope2009
Suppose that p and q are odd primes and p=2q+1. Suppose that
α∈ Z_p^*,α≢±1 mod p.

Prove that α is primitive element modulo p if and only if α^q≡-1 mod q.

That sounds like a homework problem?? Do you think?

I don't get any response on this. But since q is a "Sofie Germain Prime," then q=(p-1)/2.

By Fermat's Little Theorem z^(p-1) ==1 Mod p. IF a number is a primitive root--which I take is what he means here- no extensions are mentioned; then all its powers Modulo p must differ, giving us the p-1 residue system. ;

To do this, a^(p-1)/2 ==-1 Mod p. Since if it was 1, then a would not generate all the elements.

Last edited:
robert Ihnot said:
I don't get any response on this. But since q is a "Sofie Germain Prime," then q=(p-1)/2.

By Fermat's Little Theorem z^(p-1) ==1 Mod p. IF a number is a primitive root--which I take is what he means here- no extensions are mentioned; then all its powers Modulo p must differ, giving us the p-1 residue system. ;

To do this, a^(p-1)/2 ==-1 Mod p. Since if it was 1, then a would not generate all the elements.

thanks a lot i really appreciate your help

hope2009: thanks a lot i really appreciate your help

Happy to hear you are satisifed. However, there is an error I see now:

Prove that α is primitive element modulo p if and only if α^q≡-1 mod q.

You mean $$a^q \equiv -1 \bmod p$$ Since by Fermat's little theorem for a prime, $$a^q \equiv a \bmod q$$

Last edited:

## 1. What is a primitive element modulo p?

A primitive element modulo p is an element in a finite field or ring, where p is a prime number, that generates all the other elements in the field or ring when raised to different powers. It is also known as a primitive root modulo p.

## 2. How do you find primitive elements modulo p?

The existence of primitive elements modulo p depends on the properties of the prime number p. For example, if p is a Fermat prime, then every element in the finite field will be a primitive element modulo p. In general, finding primitive elements is a complex mathematical problem that requires advanced techniques such as the index calculus algorithm.

## 3. What are the applications of primitive elements modulo p?

Primitive elements modulo p have various applications in number theory, cryptography, and error-correcting codes. They are also used in the construction of finite fields, which have practical applications in coding theory, communication systems, and cryptography.

## 4. Can a primitive element modulo p exist in non-prime fields?

No, primitive elements modulo p can only exist in finite fields where p is a prime number. This is because the order of a finite field must be a prime power for a primitive element to exist.

## 5. How are primitive elements modulo p related to the discrete logarithm problem?

Primitive elements modulo p are closely related to the discrete logarithm problem. The discrete logarithm problem is the challenge of finding the exponent of a given element in a finite field or group. The existence of primitive elements makes this problem harder to solve, especially in large fields with a high degree of complexity.

• Linear and Abstract Algebra
Replies
2
Views
1K
• Linear and Abstract Algebra
Replies
2
Views
956
• Linear and Abstract Algebra
Replies
3
Views
1K
• General Math
Replies
5
Views
1K
• Linear and Abstract Algebra
Replies
11
Views
1K
• Calculus and Beyond Homework Help
Replies
30
Views
2K
• Linear and Abstract Algebra
Replies
3
Views
913
• General Math
Replies
11
Views
946
• Calculus and Beyond Homework Help
Replies
15
Views
2K
• Linear and Abstract Algebra
Replies
17
Views
4K