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Polynomial/Field problem

  1. Nov 17, 2004 #1
    Given a field E which contains a field F. Let f and g be polynomials with coefficients from F. And let a be an element of E such that a is a root of both f and g. Prove that there exists a polynomial h with coefficients in F that divides both f and g.
     
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  3. Nov 18, 2004 #2

    Hurkyl

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    So, what you're looking for is a common factor of f and g, right?
     
  4. Nov 19, 2004 #3

    AKG

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    To get you started:

    Case 1: a is in F, then h(t) = (t - a) divides both f and g, and it clearly takes coefficients from F.

    Case 2: a is not in F. Now f(t) and g(t) have a unique factorization in the form:

    [tex]f(t) = c(\phi _1 (t))^{n_1}(\phi _2 (t))^{n_2} \dots (\phi _k (t))^{n_k}[/tex]

    Where [itex]c[/itex] is some element of F, [itex]\forall \ i,\ \phi _i[/itex] is a monic irreducible polynomial, and [itex]n_i \in \mathbb{N}[/itex]. We also know that for some i, let it arbitrarily be 1, [itex]\phi _1 (t) = (t - a)[/itex]. For simplicity sake, let c = 1 (and something should be similarly true for [itex]g(t)[/itex]). So:

    [tex]f(t) = (t - a)^{n_1}(\phi _2 (t))^{n_2} \dots (\phi _k (t))^{n_k}[/tex]

    [tex]g(t) = (t - a)^{m_1}(\psi _2 (t))^{m_2} \dots (\psi _l (t))^{m_l}[/tex]

    You need to show that if f and g are to have only coefficients in F, then for some i and j such that [itex]1 \leq i \leq k[/itex] and [itex] 1 \leq j \leq l[/itex], [itex]\phi _i = \psi _j[/itex]. This will reduce to showing that any polynomial with (t - a) as a factor will have to have some factor with coefficients in F for the whole thing to have coefficients in F. We can write:

    [tex]f(t) = (t - a)(t^N + a_{N - 1}t^{N - 1} + \dots + a_0)[/tex]

    [tex]= t^{N + 1} + (a_{N - 1} - a)t^N + (a_{N - 2} - a_{N - 1}a)t^{N - 1} + \dots + (a_0 - a_1a)t + a_0a[/tex]

    Note that N must be at least 1, otherwise if N were 0, you'd have f(t) being (t - a) times a non-zero constant, C, and Ct - Ca = f(t), but if Ca is in F, then C is not in F (recall that a was assumed (the whole point of the second case) to not be in F, so it's not zero, so it's inverse exists and is in E but not in F (otherwise if 1/a was in F, then since F is a field, 1/(1/a) = a would be in F, contradiction), so the coefficient of "t" is C which is not in F, so N really must be at least one).

    At this point, I would try doing some induction on N to find out what form the polynomial [itex](t^N + a_{N - 1}t^{N - 1} + \dots + a_0)[/itex] must take in order for [itex](a_{N - 1} - a),\ (a_{N - 2} - a_{N - 1}a),\ \dots ,\ (a_0 - a_1a),\ a_0a[/itex] are in F. Then show that all polynomials f and g such that [itex](t^N + a_{N - 1}t^{N - 1} + \dots + a_0)[/itex] takes the required form must be factorizable by some polynomial h over F.

    Start with n = 1. [itex]f(t) = (t - a)(t + a_0) = t^2 + (a_0 - a)t - a_0a[/itex]. Now [itex]a \in E,\ a \notin F,[itex] so [/itex]a \neq 0[/itex], but [itex]a_0a \in F[/itex], so [itex]a_0 = k/a[/itex] for some [itex]k /in E[/itex]. If k = 0, then the coefficient of "t" is not in F, so [itex]k \neq 0[/itex]. Now, we have:

    [tex]a_0 - a = k/a - a \in F[/tex]

    You want to convince yourself that the k is unique. Assume for some other [itex]K \in F,\ K \neq k[/itex] where clearly, [itex]K \neq 0[/itex], we have that [itex]K/a - a \in F[/itex]. Then:

    [tex]K/a - a - (k/a - a) \in F[/tex]

    [tex]K/a - k/a \in F[/tex]

    [tex](1/a)(K - k) \in F[/tex]

    [itex]K - k = La[/itex] for some [itex]L \in F[/itex]. Now if [itex]L \neq 0[/itex], then [itex]K = k + La[/itex] but [itex]k \in F[/itex], [itex]La \notin F[/itex], so [itex]k + La \notin F[/itex] while [itex]K \in F[/itex], contradiction, so [itex]L = 0[/itex], and hence [itex]K = k[/itex], so k is unique. Now, if you can show that every polynomial over F that has (t - a) as a factor must also have (t + k/a) as a factor, then you know that every such polynomial will have (t - a)(t + k/a) = t² + (k/a - a)t - k as it's factor, and h(t) = t² + (k/a - a)t - k is a polynomial over F.

    I hope this is on the right track. I would hope you know some theorems that you can use to do this problem, I'm working with no special knowledge of the subject.
     
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