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Math Help - 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) F_1=\mathbb{Z}[i]/<3>
    ii) F_2=\mathbb{R}[x]/<x^2+1>
    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]/<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
    Posts
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    For (ii)

    You can prove that \mathbb R[x]/ <x^2+1>  \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.
    Last edited by zoek; April 9th 2011 at 07:36 AM.
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