# Math Help - Characteristics and prime fields

1. ## Characteristics and prime fields

Hello,
the task is to define characteristics and prime fields of following fields:
i) $F_1=\mathbb{Z}[i]/<3>$
ii) $F_2=\mathbb{R}[x]/$
iii) $F_3=\mathbb{Z}[i]/<1+i>$

The $F_1$ seems to be following kind of field:
$\mathbb{Z}[i]/<3>=\{a+bi+<3>| a,b \in \mathbb{Z} \}=\{a+bi+<3>|a,b \in \{0,1,2 \} \}$.
So, the $F_1$'s order is $9$, but I don't understand, what is it's characteristic. Prime field should be something like:
$\{<3>, 1+<3>, 2+<3> \}$.

I think the second one is: ${\mathbb{R}[x]/= \{p(x)+|p(x) \in \mathbb{R}\}= \{a_0+a_1x+| a_i \in \mathbb{R} \}$.
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 $\mathbb R[x]/ \cong \mathbb C$, so its characteristic is $0$ and its prime field is $\mathbb Q$.

For (iii)

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

for each element we have:

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

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