Hello,

the task is to define characteristics and prime fields of following fields:

i) $\displaystyle F_1=\mathbb{Z}[i]/<3>$

ii) $\displaystyle F_2=\mathbb{R}[x]/<x^2+1>$

iii) $\displaystyle F_3=\mathbb{Z}[i]/<1+i>$

The $\displaystyle F_1$ seems to be following kind of field:

$\displaystyle \mathbb{Z}[i]/<3>=\{a+bi+<3>| a,b \in \mathbb{Z} \}=\{a+bi+<3>|a,b \in \{0,1,2 \} \} $.

So, the $\displaystyle F_1$'s order is $\displaystyle 9$, but I don't understand, what is it's characteristic. Prime field should be something like:

$\displaystyle \{<3>, 1+<3>, 2+<3> \}$.

I think the second one is: $\displaystyle {\mathbb{R}[x]/<x^2+1>= \{p(x)+<x^2+1>|p(x) \in \mathbb{R}\}= \{a_0+a_1x+<x^2+1>| a_i \in \mathbb{R} \}$.

But I have no idea, what it's characteristic or prime field could be.

The third one is completely mysterious to me - I don't understand it's structure at all.

Any help is appreciated. Thanks very much in advance!