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Math Help - What is the dimension of this vector space

  1. #1
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    What is the dimension of this vector space

    Let p(t) be a polynomial of degree n, in the F[t] polynomial ring, where F is a field. Let be a congruence relation for: r(t) s(t) <==> p(t)|r(t) - s(t).

    What is the dimension of the vector space whose elements are the elements of F[t]/. What is the basis of the vector space?
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  2. #2
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    Re: What is the dimension of this vector space

    well, clearly, every multiple of p(t) is congruent to the 0-polynomial. and since we can write, for ANY polynomial f(t) in F[t]:

    f(t) = p(t)q(t) + r(t), where the degree of r is less than the degree of p, it's clear that every element of F[t]/β can be written:

    r(t) + B, where B is the equivalence class of 0 (and thus p(t)) under β. and since r has degree n-1 or less:

    \{1+B, t+B, t^2+B,....,t^{n-1}+B\} is a spanning set for F[t]/β. so dim(F[t]/β) ≤ n.

    now suppose c_0(1+B) + c_1(t+B) +\dots+c_{n-1}(t^{n-1}+B)

     = (c_0 + c_1t +\dots+c_{n-1}t^{n-1}) + B = 0 +B

    this implies p(t) divides c_0 + c_1t +\dots+c_{n-1}t^{n-1},

    and since the latter polynomial has degree < degree p(t), it must be the 0-polynomial. that is:

    c_0 = c_1 = \dots = c_{n-1} = 0, so dim(F[t]/β) ≥ n. that leaves just one possibility....
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  3. #3
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    Re: What is the dimension of this vector space

    Thank you!
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