Multiplication of primitive roots

In summary, the conversation discusses the multiplication of all primitive roots modulo p for p > 3, which is congruent to 1 modulo p. The speaker has tried some examples but cannot prove the general case for g1...gk as primitive roots modulo p. They are seeking help to either prove or disprove this statement. The other participant suggests multiplying the roots in a clever order and leaving no stone unturned. The speaker also asks for an explanation on why the inverse of a primitive root is also primitive. The other participant mentions that it is a group and the order of x and x^-1 are the same.
  • #1
trisacloon
3
0
Hi

I noticed that multiplication of all primitive roots modulo p ,p>3,
congruent to 1 modulo p...

I have tried some examples (13,17,19...) but i couldn't prove the general case
(let g1...gk be primitive roots modulo p,p>3 ==> g1*g2*...*gk=1(p))

I need help to prove or disprove it...

thanks in advance
 
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  • #2
trisacloon said:
Hi

I noticed that multiplication of all primitive roots modulo p ,p>3,
congruent to 1 modulo p...

I have tried some examples (13,17,19...) but i couldn't prove the general case
(let g1...gk be primitive roots modulo p,p>3 ==> g1*g2*...*gk=1(p))

I need help to prove or disprove it...

thanks in advance
Try multiplying them in a clever order.
 
  • #3
thanks for the replay
I'll try to follow your lead ...
 
  • #4
I would suggest looking at detail, leave no stone unturned!
 
  • #5
I will rephrase my question...
How can i explain that if g is a primitive root then g^-1 is primitive root?
 
  • #6
trisacloon said:
I will rephrase my question...
How can i explain that if g is a primitive root then g^-1 is primitive root?
I would have expected it to be straightforward, although I haven't worked it out explicitly.

Where, when proving the inverse of a primitive root is primitive, do you get stuck?
 
Last edited:
  • #7
It's a group. The order x and x^-1 are trivially shown to be the same.
 
  • #8
If 1/g^k ==1, Mod M, then of course, g^k==1 Mod M.
 

1. What are primitive roots?

Primitive roots are numbers that, when raised to certain powers, generate all the elements in a finite field. They are also known as generators or primitive elements.

2. How do you multiply primitive roots?

To multiply primitive roots, you can use the property that the product of two primitive roots is another primitive root. So if a and b are primitive roots, then a*b is also a primitive root.

3. What is the significance of multiplication of primitive roots?

Multiplication of primitive roots is important in the study of finite fields, as it helps to generate all the elements in the field. It is also used in cryptography and coding theory.

4. Can you have more than one primitive root in a finite field?

Yes, a finite field can have multiple primitive roots. In fact, the number of primitive roots in a finite field is equal to the number of elements in the field minus one.

5. How is multiplication of primitive roots related to modular arithmetic?

Multiplication of primitive roots is closely related to modular arithmetic. In fact, the powers of a primitive root can be thought of as the remainders obtained when dividing by the order of the finite field. This relationship is often used in computations involving primitive roots.

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