1. ## Irreducible Polynomials

a) What is the percent likelihood that a random polynomial over $\displaystyle F_2$ 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 $\displaystyle F_3$ 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?

2. Originally Posted by lttlbbygurl
a) What is the percent likelihood that a random polynomial over $\displaystyle F_2$ of degree exactly 10 factors into a product of polynomials of degree less than or equal to 2?
There are $\displaystyle 2^{10}$ 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 ($\displaystyle x$ and $\displaystyle x+1$) and one irreducible quadratic factor ($\displaystyle x^2+x+1$). For a polynomial of degree 10 with k copies of the quadratic factor (k=0,1,2,3,4 or 5), there are 11–2k ways of choosing the remaining linear factors (for j=0,1,2,...,10–2k 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.