What is the Probability of Factoring Random Monic Polynomials?

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SUMMARY

The probability of a random monic polynomial over F_3 of degree exactly 10 factoring into polynomials of degree less than or equal to 2 is a complex combinatorial problem. The discussion highlights the need to calculate the total number of monic polynomials of degree 10, as well as the counts of monic polynomials of degrees 1 and 2. Specifically, it emphasizes the importance of understanding the structure of irreducible polynomials in this context to derive accurate probabilities.

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  • Understanding of monic polynomials
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I've been trying to work out a bunch of problems that have to do with finding irreducible polynomials, and this one really seemed to stump me...

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?
 
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Let's do monic polynomials. (Multiply them by 2 to get the other polynomials, and similarly for the other factorizations, so the probability answer is the same.)

How many different monic polynomials of degree exactly 10 are there?

How many different monic polynomials of degree exactly 1?
How many different irreducible monic polynomials of degree exactly 2?

How many different ways can you multiply these to get a polynomial of degree exactly 10?
 

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