2

My English is not very good, and that's why I would really appreciate it if you could explain to me what the phrase : these elements are under the same domain under $F$ and $\alpha$ means in this article: http://www.pagewizardgames.com/finitefields_primeorder_danielpage.pdf.

Could you help me with that?

By the way, do you know a fairly simple proof of the theorem stating that for any integers $m, \ n, \ r > 1$, there exists a finite group $G$ with elements $a$ and $b$ such that $a$ has order $m$, $b$ has order $n$, and $ab$ has order $r$?

Sven
  • 45
  • 5
    Your second question should be asked separately (I think it is already somewhere on the site though). – Qiaochu Yuan Apr 24 '13 at 18:51
  • Ok, I'll try looking for it. What about the first one? – Sven Apr 24 '13 at 18:53
  • 4
    The linked document employs neither standard mathematical language nor rigorous mathematical proof. Instead, see any Abstract Algebra textbook for a proof of Cauchy's Theorem. – Math Gems Apr 24 '13 at 19:04
  • 2
    I think the construction answering the second question uses contrived $\operatorname{SL}$ groups. – Alexander Gruber Apr 24 '13 at 19:57
  • Yes, I've just found it. It's here: http://www.jmilne.org/math/CourseNotes/GT310.pdf page 28. Do you think you could answer one question I have about the proof? – Sven Apr 24 '13 at 20:02
  • Namely, how do we know that if $p$ be a prime number not dividing $2mnr$, $p$ is a unit in $\mathbb{Z}/2mnr \mathbb{Z}$, and $q=p^k, \ k \in \mathbb{Z}$ is the ring's identity, then $2mnr$ divides $q -1$ and there exist elements $a, \ b, \ c$ with orders $2m, \ 2n, 2r$ – Sven Apr 24 '13 at 20:07
  • 2
    I've already posted it as a new question. http://math.stackexchange.com/questions/371851/ring-mathbbz-2mnr-mathbbz-unit-identity-orders – Sven Apr 24 '13 at 20:19

1 Answers1

2

I am finding the linked page a little hard to read myself. Without reading more intensely to make sure, I have the feeling the author is not very clear on the topic.

At any rate, if you want a proof that finite fields have prime power orders, there are much clearer and simpler explanations than that, such as this question:

Order of finite fields is $p^n$

Community
  • 1
rschwieb
  • 153,510