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Thread: Characteristics and prime fields

  1. #1
    Junior Member Greg98's Avatar
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    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!
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  2. #2
    Junior Member
    Joined
    Mar 2011
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    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 $\displaystyle \mathbb Z_2$.
    Last edited by zoek; Apr 9th 2011 at 07:36 AM.
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