5
$\begingroup$

Is there a finite field whose additive group is not cyclic?

  • 4
    For any prime power $q=p^r$ (here $p$ is prime), $\Bbb F_{q}\cong\Bbb F_p[x]/(x^q-x)$, hence $(\Bbb F_q,+)\cong C_p^r$. For $r>1$ this is not cyclic.2012-08-25
  • 6
    I’ve found that understanding proceeds from the knowledge of examples. Start with the field ${\mathbb{Z}}/(3)$ with three elements, and formally adjoin the square root of $-1$, i.e. the square root of $2$. Play around in it, splash and frolic, add and multiply, look for an element whose powers exhaust all nonzero elements. You will then know much more than before!2012-08-25
  • 0
    @anon what is $C_p^r$ ? Is it is. $Z_p × Z_p × ...× Z_p$ (r times) ?2017-03-24

1 Answers 1