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Thread: 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 $\displaystyle \{a+bi \ | \ a,b \in F_3 \}$ 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 $\displaystyle F_5$? For $\displaystyle F_7$? Explain.

    So I can see that the nine elements are: $\displaystyle \{0,1,2,i,1+i,2+i,2i,1+2i,2+2i \}$ 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 $\displaystyle \mathcal{S}_p$ of symbols to form a field. For example, taking into account that $\displaystyle F_p=\{0,1,\ldots, p-1\}$ with $\displaystyle p$ prime is a field, then. given $\displaystyle a+ib\in\mathcal{S}_p$ then,

    $\displaystyle (a+ib)+(-a+(-b)i)=0+0i$

    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:
    $\displaystyle (a+bi)+(c+di)=(a+c)+i(b+d)$ and $\displaystyle (a+bi)(c+di)=(ac-bd)+i(ad+bc)$

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

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

    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; Sep 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 $\displaystyle F_5$ and $\displaystyle F_7$, 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 $\displaystyle \{a+bi \ | \ a,b \in F_5 \}$ and $\displaystyle \{a+bi \ | \ a,b \in F_7 \}$ to see if they form a field, if the laws of composition are made to mimic addition and multiplication of complex numbers.

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

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