# How to construct a specific isomorphism between two finite fields with same order?

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?

• 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. Commented Oct 15, 2015 at 21:55

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 $(2\sqrt{-3})^2=-1$.