Estimate proportion of polynomial

Question

I need:

Estimate the proportion of polynomial of degree $n$ in $\mathbb{F}_q[x]$, with a large $q$ that are irreducible

Estimate the proportion of polynomial of degree $n$ that descompose totally over $\mathbb{F}_q$.

So, my first problem is what you refer by "proportion", I though write the expression of the form, for instance, $q^n+\mathcal{O}(q^{n/2})$.

On the other hand, I know that the number $N_{q}(n)$ of monic irreducible polynomials in $\mathbb{F}_{q}[x]$ of degree $n$ is given by $$N_q(n)=\dfrac{1}{n}\sum_{d\mid n}{\mu(d)q^{\frac{n}{d}}}$$ But anyway, I don't know how estimate a proportion of this polynomial over $\mathbb{F}_{q}[x]$. Can help me pls! Thanks!

See http://math.stackexchange.com/questions/848146/number-of-monic-irreducible-polynomials-over-a-finite-field — Gerry Myerson, Dec 13 '16 at 06:21

score 0 · Accepted Answer · answered Dec 13 '16 at 05:31

For the first, there are $q^{n+1}$ polynomials of degree $n$ because each coefficient has $q$ choices. You are supposed to estimate the number of these that are reducible, then divide by $q^{n+1}$ to get the proportion. The degree of one factor has to be less than or equal to $n/2\ \ldots$

For the second, you need to estimate the number of products of first degree polynomials. You need to think about rearrangements of factors producing the same ultimate polynomial.

Estimate proportion of polynomial

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