Order of element

  • #1
Is there a term for the least k such that a^k = 1 in some ring R (if it exists, specifically in a finite ring)? Is there a way to get an upper bound in general? I know that it's hard to get even in say F_p, but I'm just looking for some conditions that would help me understand what it looks like.
 

Answers and Replies

  • #2
22,089
3,293
Is there a term for the least k such that a^k = 1 in some ring R (if it exists, specifically in a finite ring)?
The term order should work well.

Is there a way to get an upper bound in general?
Let ##n## be the number of invertible elements in the ring, then ##n## is an upper bound.

I know that it's hard to get even in say F_p
Well, you know that the invertible elements in ##\mathbb{F}_p## form a cyclic group of order ##p-1##. So finding the orders of the elements in equivalent to finding the orders of elements in ##\mathbb{Z}_{p-1}##, which is easy. So you know exactly which orders show up and which don't. The hard part is finding an explicit isomorphism between ##\mathbb{Z}_{p-1}## and the invertible elements in ##\mathbb{F}_p## (or equivalently: finding an element in ##\mathbb{F}_p## with order ##p-1##).
 

Related Threads on Order of element

  • Last Post
Replies
2
Views
16K
  • Last Post
Replies
13
Views
264
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
10
Views
6K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
5
Views
8K
Replies
4
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
2K
Top