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Math Help - Algebra help: Find a primitive root β of F3[x]/(x^2 + 1)...etc

  1. #1
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    Algebra help: Find a primitive root β of F3[x]/(x^2 + 1)...etc

    (a) Find a primitive root β of F3[x]/(x^2 + 1).
    (b) Find the minimal polynomial p(x) of β in F3[x].
    (c) Show that F3[x]/(x^2 + 1) is isomorphic to F3[x]/(p(x)).

    Please help, im lost on this one.
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  2. #2
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    Quote Originally Posted by habsfan31 View Post
    (a) Find a primitive root β of F3[x]/(x^2 + 1).
    (b) Find the minimal polynomial p(x) of β in F3[x].
    (c) Show that F3[x]/(x^2 + 1) is isomorphic to F3[x]/(p(x)).

    Please help, im lost on this one.

    If you meant \mathbb{F}_3[x]= the ring of pol's over the field with three elements, then \mathbb{F}_3[x]/(x^2+1) is a

    field with 9 elements (why?) , so \beta is a generator of the multiplicative group of this field. What have you tried/done?

    Tonio
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  3. #3
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    I havent really tried anything, im pretty lost when it comes to this topic and the book is really confusing. I already handed in the assignment with this question blank but would still like to know how to solve it. Can you possibly show me how its done step by step?
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  4. #4
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    Can you tell me please how does the elements in F_3[x]/ (x^2+1) look like in order to be able to construct the addition and multiplication tables
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  5. #5
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    Quote Originally Posted by Mike12 View Post
    Can you tell me please how does the elements in F_3[x]/ (x^2+1) look like in order to be able to construct the addition and multiplication tables

    Suppose w is an element in some extension of \mathbb{F}_3 s.t. w^2+1=0\Longleftrightarrow w^2=-1 , so every

    element in \mathbb{F}_3[x]/<x^2+1> is of the form aw+b , with a,b\in\mathbb{F}_3 and

    sum and multiplication modulo 3, taking into account that w^2=-1.

    Thus, for example, w\cdot (2w+1)=2w^2+w=(-2)+w=1+w , etc.

    Tonio
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  6. #6
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    can you tell me please how to prove that f(x)=x^5+x^4+x^3+x^2+x+1 belongs to F_2[x] has a multiple factor. and how can I find the all factors of f(x) and their multiplicities
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  7. #7
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    Quote Originally Posted by Mike12 View Post
    can you tell me please how to prove that f(x)=x^5+x^4+x^3+x^2+x+1 belongs to F_2[x] has a multiple factor. and how can I find the all factors of f(x) and their multiplicities

    For any natural number n , x^n-1=(x-1)(x^{n-1}+x^{n-2}+\ldots+x+1)

    By "multiple factor" I understand that you meant that the polynomial factors...right?!

    Tonio
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  8. #8
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    thanks for replying , yes I meant finding the all factors
    please , help me
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  9. #9
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    This concerns me. You seem to know nothing about what, if you know the basic definitions, is a fairly simple problem just involving "arithmetic". Are you clear on what " F_2" is?

    In F_2, the only "numbers" are 0 and 1: 0+ 0= 1, 0+1= 1+0= 1, 1+1= 0; 0*0= 0*1= 1*0= 0, 1*1= 1. If x= 0, then f(0)=0+ 0+ 0+ 0+ 0+ 1 but if x= 1, then f(1)= 1+1+ 1+ 1+ 1+ 1= 0 just because there are an even number of terms. That's how we know x-1 is a factor and, by synthetic division, that it is equal to (x- 1)(x^4+ x^2+ 1) Now, if g(x)= x^4+ x^2+ 1, then g(0)= 0+0+ 1= 1 and g(1)= 1+ 1+ 1= 1 so g has no linear factors and so no third degree factors. The only possible way to factor g would be as two quadratic factors and the only possible "quadratic" polynomials in F_2(x) are of the form x^2+ ax+ b with a and b either 0 or 1. That is, there are only four possible quadratics: x^2, x^2+ 1, x^2+ x, and x^2+ x+ 1. You could try dividing x^4+ x^2+ 1 by those (being careful to use " F_2" arithmetic) to see if it factors and, if so, what those factors are.
    Last edited by HallsofIvy; January 27th 2011 at 08:19 AM.
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