2
$\begingroup$

In field theory we know that if two finite fields have the same order $P^n$, then they are isomorphic and can be identified with the field $ \mathbb F_p$. However, I encountered some problem when dealing with some specific fields. Consider $ \mathbb F_{11}[\sqrt -1]$ and $F_{11}[\sqrt -3]$. Both are fields of order 121 and they are isomorphic. However, I find it hard to give an explicit isomorphism between this two fields. Can anyone help me with this question?

$\endgroup$
1
  • $\begingroup$ The difficulty is from the fact that such finite fields have non-trivial Galois groups over their prime fields, so that there may be no canonical isomorphism, and certainly not a unique one, since one can always tweak it by an automorphism. $\endgroup$ Commented Oct 15, 2015 at 21:55

1 Answer 1

2
$\begingroup$

Since both fields the polynomial, $x^{121}-x$, has $121$ roots in both fields, and $x^2+1$ is a factor of that polynomial in $\mathbb Z_{11}[x]$, we know that $x^2+1$ splits in $\mathbb F_{11}[\sqrt{-3}]$. Just find a root of that in the second field, and you can send $\sqrt{-1}$ to that element. (Okay, finding the root is possibly non-trivial in general, but we know it can be done.)

In your particular case, though, $\sqrt{-3}=\pm 5\sqrt{-1}$, because $5^2=3\pmod {11}$. So $(2\sqrt{-3})^2=-1$.

$\endgroup$
2
  • $\begingroup$ I took a quick look at the Wiki article Finite Fields. The sections "Existence and Uniqueness" and "Explicit Construction" or the in-line references in them may be helpful to you. $\endgroup$ Commented Oct 15, 2015 at 21:58
  • $\begingroup$ Ahh! That is very nice. I tried some other examples and re-read the sections in my book and fully understand it. Thanks a lot! $\endgroup$
    – user194201
    Commented Oct 15, 2015 at 22:05

You must log in to answer this question.