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Math Help - Rings and fields - Write down the nine elements of F9[i] help

  1. #1
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    Rings and fields - Write down the nine elements of F9[i] help

    In F9 = Z/3Z, there is no solution of the equation x^2 = −1, just as in R. So “invent”
    a solution, call it 'i'. Then 'i' is a new “number” which satisfies i^2 = −1. Consider
    the set F9[i] consisting of all numbers a+bi, with a,b in F9. Add and multiply these
    numbers as though they were polynomials in 'i', except whenever you get i^2 replace
    it by −1.
    (i) Write down the nine elements of F9[i] .
    (ii) Show that every nonzero element of F9[i] has an inverse, so that F9[i] is a
    field.

    I know im supposed to show you that ive tried the question if i want an answer. Believe me, i have tried it. Im just really confused by the wording of the question and am not really sure what they are looking for in part a. Once i get part a, im pretty sure id be able to get part b on my own.

    Thanks
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  2. #2
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    Quote Originally Posted by habsfan31 View Post
    In F9 = Z/3Z, there is no solution of the equation x^2 = −1, just as in R. So “invent”
    a solution, call it 'i'. Then 'i' is a new “number” which satisfies i^2 = −1. Consider
    the set F9[i] consisting of all numbers a+bi, with a,b in F9. Add and multiply these
    numbers as though they were polynomials in 'i', except whenever you get i^2 replace
    it by −1.
    (i) Write down the nine elements of F9[i] .
    (ii) Show that every nonzero element of F9[i] has an inverse, so that F9[i] is a
    field.

    I know im supposed to show you that ive tried the question if i want an answer. Believe me, i have tried it. Im just really confused by the wording of the question and am not really sure what they are looking for in part a. Once i get part a, im pretty sure id be able to get part b on my own.

    Thanks

    Say \mathbb{F}_3:\{0,1,2\} , with operations modulo 3, so F_3[i]:=\{a+bi / a,\,b\in \mathbb{F}_3\} , with operations

    modulo 3 and also taking into account that i^2=-1=2\!\!\pmod 3 .

    1) Show first that indeed |\mathbb{F}_3[i]|=9 ;

    2) Show that all the axioms of field are fulfilled in \mathbb{F}_3[i] , in particular identify the neutral

    elements wrt addition and multiplication (it's basically the same as with the "usual" complex numbers)

    3) Now show that every non-zero element has an inverse (again, \mathbb{C} is a good example)

    Tonio
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