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Math Help - Irreducible Polynomials

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    Irreducible Polynomials

    a) What is the percent likelihood that a random polynomial over  F_2 of degree exactly 10 factors into a product of polynomials of degree less than or equal to 2? What is the likelihood that a random nonzero polynomial of degree at most 10 factors into such a product?

    (b) What is the probability that a random monic polynomial over  F_3  of degree exactly 10 factors into a product of polynomials of degree less than or equal to 2? What is the probability that a random monic polynomial of degree at most 10 factors into such a product?
    Last edited by lttlbbygurl; July 23rd 2009 at 11:33 PM.
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  2. #2
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    Quote Originally Posted by lttlbbygurl View Post
    a) What is the percent likelihood that a random polynomial over  F_2 of degree exactly 10 factors into a product of polynomials of degree less than or equal to 2?
    There are 2^{10} polynomials of degree 10 (for each power of x from 0 to 9 inclusive, there are two possible choices of coefficient, namely 0 or 1).

    So, how many of these can be factored into a product of linear or quadratic factors? There are two possible linear factors ( x and x+1) and one irreducible quadratic factor ( x^2+x+1). For a polynomial of degree 10 with k copies of the quadratic factor (k=0,1,2,3,4 or 5), there are 112k ways of choosing the remaining linear factors (for j=0,1,2,...,102k you can choose j of them to be the factor x and the remaining ones to be the factor x+1).

    That should give you enough information to calculate the proportion of degree 10 polynomials that have the required form of factorisation.
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