0
$\begingroup$

How to decide if the multiplicative group of non-zero elements of a finite field is cyclic or not?

Based on our experience with $\left(\mathbb{Z}_7 \setminus \{0\}, \cdot \right)$, which is generated by $3$ or $5$, and with $\left(\mathbb{Z}_{11} \setminus \{0\}, \cdot\right)$, which is generated by $2$, $6$, $7$, or $8$, can we conclude that $\left(\mathbb{Z}_p \setminus \{0\}, \cdot\right)$, where $p$ is any prime, is cyclic? And if so, then which elements can generate this group?

$\endgroup$
  • 3
    $\begingroup$ Pretty much every textbook dealing in any detail with finite fields will prove that the multiplciative group is cyclic. Have you tried looking in one? (In fact, every finite subgroup of the multiplicative group of a field is automatically cyclic, and this implies the result you want) $\endgroup$ – Mariano Suárez-Álvarez Oct 21 '13 at 6:00
  • 3
    $\begingroup$ As for your second question, and restricting myself to fields of prime order, have you read en.wikipedia.org/wiki/Primitive_root_modulo_n ? $\endgroup$ – Mariano Suárez-Álvarez Oct 21 '13 at 6:02
  • 1
    $\begingroup$ The firs google hit for "finite fields cyclic": mathoverflow.net/questions/54735/…. You're welcome :) $\endgroup$ – Prahlad Vaidyanathan Oct 21 '13 at 6:07
  • $\begingroup$ Mariano Suárez-Alvarez, I'm going through an abstract algebra text in sequence and have so far reached a discussion of rings and integral domains. $\endgroup$ – Saaqib Mahmood Oct 21 '13 at 7:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.