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Homework Help: Please explain this question to me (F2 field)

  1. Sep 25, 2010 #1

    Suy

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    Please explain these question to me ....

    1. The problem statement, all variables and given/known data

    1)A system of m equations over F2 in n
    unknowns has exactly 2n−m solutions.
    • A. True
    • B. False
    answer: false

    2)How many roots x [tex]\in[/tex] F2 does the polynomial
    x2+x+1 have?
    answer:0

    3)Which of the following is a cube root of 8i? (Choose all correct answer)
    A. sqrt(3)−i
    B. −sqrt(3)−i
    C. sqrt(3)+i
    D. −2i
    E. −sqrt(3)+i
    answer: sqrt(3)+i( i think there is more?)

    2. Relevant equations



    3. The attempt at a solution
    I have the answer, but I don't understand
    also, those stuff is not in my textbook and the prof didn't give much example
    .... please help
     
    Last edited: Sep 25, 2010
  2. jcsd
  3. Sep 25, 2010 #2

    Office_Shredder

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    Re: Please explain this question to me .... (F2 field)

    Let's start the second and third questions, since they're more straightforward. For b, there are only two possible roots that you can check don't work. Do you know how to do this?

    For c, can you just check whether each option is a cube root of 8i directly?
     
  4. Sep 25, 2010 #3

    Suy

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    Re: Please explain this question to me .... (F2 field)

    "b" you mean question 2?
    No ,I have no clue.... I don't know to cube root i...
    The third question answer is come from wolfram..
    By checking, Do you mean i cube every answer to see if it is equal 8i?
    actually,I don't understand what's F2 is ...
     
    Last edited: Sep 25, 2010
  5. Sep 25, 2010 #4

    Office_Shredder

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    Re: Please explain this question to me .... (F2 field)

    Right, by b I meant the second question and c the third question

    F2 is just the field of two elements. It's the same thing as [tex]\mathbb{Z}/2\mathbb{Z}[/tex], the integers modulo 2.

    If x is a cube root of 8i, then x3=8i. So can you check whether the five options given are cube roots of 8i?
     
  6. Sep 25, 2010 #5

    Suy

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    Re: Please explain this question to me .... (F2 field)

    answer is C,D,E, but is there a easy way to do it??
    for question 2, I know there is no solution, but the question say x [tex]\in[/tex] F2 confused me
    And what about question 1 please help me

    edit:

    If z/2z, is that mean (x+iy)/(2(x+iy))?
    Thank you so much!
     
    Last edited: Sep 26, 2010
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