# Thread: Characteristics and prime fields

1. ## Characteristics and prime fields

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!

2. For (ii)

You can prove that $\displaystyle \mathbb R[x]/ <x^2+1> \cong \mathbb C$, so its characteristic is $\displaystyle 0$ and its prime field is $\displaystyle \mathbb Q$.

For (iii)

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

for each element we have:

$\displaystyle 2\cdot \left( a+bi + <1+i> \right) = 2\cdot \left(a+bi \right)+<1+i> = 2a+2bi+<1+i> = \left[a+b+\left(b-a \right)i \right]\left(1+i \right)+<1+i> = <1+i>$

so, the characteristic of this field is $\displaystyle 2$ and its prime field is $\displaystyle \mathbb Z_2$.