Number of primative roots in finite fields of order p^n

In summary, a primitive root is an element in a finite field of order p^n that generates all other elements in the field when raised to different powers. The number of primitive roots in a finite field of order p^n is equal to the Euler totient function of p^n, denoted as φ(p^n). Not every element in a finite field of order p^n can be a primitive root, as there are only φ(p^n - 1) primitive roots in the field. To find primitive roots, we can use the formula g^k mod p^n, where g is any element in the field and k ranges from 1 to p^n - 1. Primitive roots are important in finite fields because they allow for efficient generation of
  • #1
Bourbaki1123
326
0
Is it true as it is for finite fields of order p^1, that the number of primitive roots of fields of order p^n is the euler totient of (P^n-1)? If not is there a different rule for the number?
 
Physics news on Phys.org
  • #2
Well, say you have a finite field F = GF(pn). Then F*, the multiplicative group of units of F, is cyclic and has order pn - 1. But the number of generators of a cyclic group G is φ(|G|), so F* has φ(pn - 1) generators, i.e. F has φ(pn - 1) primitive elements.

Is that what you were asking?
 

Similar threads

I Finite fields, irreducible polynomial and minimal polynomial theorem
Replies
6
Views
2K
I Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
Replies
0
Views
504
I Question on Primitive Roots and GCDs
Replies
2
Views
2K
Definition of Finite Field: Addition and Multiplication Groups
Replies
1
Views
2K
I Existence and Uniqeness of Finite Fields ....
Replies
3
Views
2K
I Generators of Galois group of ## X^n - \theta ##
Replies
3
Views
2K
MHB What Is the Upper Bound of Groups of Order in Finite Group Theory?
Replies
2
Views
1K
MHB Existence and Uniqeness of Finite Fields .... Example from D&F ....
Replies
2
Views
2K
MHB Field extensions and Galois goup
Replies
2
Views
2K
I Is the Order of an Automorphism in a Field with Characteristic p Equal to p?
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top