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Math Help - set of symbols form a field

  1. #1
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    set of symbols form a field

    Hi all, I am going through a Algebra textbook and I need a bit of help with this question, so thank you in advance for any help.

    Question: Prove that the set of symbols \{a+bi \ | \ a,b \in F_3 \} forms a field with nine elements, if the laws of composition are made to mimic addition and multiplication of complex numbers. Will the same method work for F_5? For F_7? Explain.

    So I can see that the nine elements are: \{0,1,2,i,1+i,2+i,2i,1+2i,2+2i \} but I am confused as to where to go from here.

    June
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: set of symbols form a field

    Well, write all the conditions for the set \mathcal{S}_p of symbols to form a field. For example, taking into account that F_p=\{0,1,\ldots, p-1\} with p prime is a field, then. given a+ib\in\mathcal{S}_p then,

    (a+ib)+(-a+(-b)i)=0+0i

    etc, etc,... . Show your work and we check it.
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  3. #3
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    Re: set of symbols form a field

    Thanks FernandoRevilla, sorry for my late reply, been gone a few days.

    So,
    Addition and Multiplication:
    (a+bi)+(c+di)=(a+c)+i(b+d) and (a+bi)(c+di)=(ac-bd)+i(ad+bc)

    Inverse:
    (a+bi)^{-1}=\frac{a-bi}{a^2+b^2}

    Distributive:
    (a+bi)[(c+di)+(e+fi)]=(a+bi)(c+di)+(a+bi)(e+fi)

    The complex addition identity is 0.
    The complex multiplication identity is 1.

    And commutative and associative follow easily.

    Is this correct working out?

    Thanks
    Last edited by Juneu436; September 24th 2011 at 05:58 PM.
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  4. #4
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    Re: set of symbols form a field

    And what about F_5 and F_7, will the same method work for these?

    Thanks
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  5. #5
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    Re: set of symbols form a field

    So I need to look at \{a+bi \ | \ a,b \in F_5 \} and \{a+bi \ | \ a,b \in F_7 \} to see if they form a field, if the laws of composition are made to mimic addition and multiplication of complex numbers.

    So for \{a+bi \ | \ a,b \in F_5 \} the elements are:
    \{0,1,2,3,4,i,2i,3i,4i,1+i,2+i,3+i,4+i,1+2i,1+3i,1  +4i,2+2i,2+3i,2+4i,3+2i,3+3i,3+4i,4+2i,4+3i,4+4i\}

    So where do I go from here?

    Any help guys?

    Thanks
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Re: set of symbols form a field

    Quote Originally Posted by Juneu436 View Post
    Inverse: (a+bi)^{-1}=\frac{a-bi}{a^2+b^2}
    Right, but you need to verify that if a+bi \neq 0 then, a^2+b^2\neq 0 . Easily proved, this condition is verified in F_3 but not in F_5 : choose for example 1+2i ( 1^2+2^2=0 ) . Try now with F_7 .
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