# Random algebraic numbers are linearly disjoint almost surely?

I already posted this question at MSE here, but since it received no answer or comment so far I cross-post it here.

It is well-known that if one considers a “random” monic polynomial of fixed degree, say $X^n+\sum_{k=0}^{n-1}a_kX^k$ where $(a_0,a_1,\ldots, a_n)$ is drawn from the discrete uniform distribution on $[-N,N]^{n+1}$, then this polynomial will be irreducible and have Galois group $S_n$ “almost surely”, i.e. the probability of this event tends to $1$ when $N\to \infty$.

Now, suppose one considers two random monic polynomials $P=X^n+\sum_{k=0}^{n-1}a_kX^k$ and $Q=X^m+\sum_{k=0}^{n-1}b_kX^k$ where $(a_0,a_1,\ldots, a_n,b_0,\ldots,b_m)$ is drawn from the discrete uniform distribution on $[-N,N]^{n+m+2}$. Is it also true that for any root $\alpha$ of $P$ and any root $\beta$ of $Q$, the extensions ${\mathbb Q}(\alpha)$ and ${\mathbb Q}(\beta)$ will be linearly disjoint over $\mathbb Q$ almost surely?

## 1 Answer

This should indeed be true and easy to prove using the large sieve approach of Gallagher ("The large sieve and probabilistic Galois theory", Proceedings of Symposia in Pure Mathematics 24, 1973, A.M.S., 91–101) -- probably most other proofs of this result should also extend. Precisely, this should give a bound of order of magnitude at most $N^{n+m-1/2}(\log N)^c$ (for some $c>0$) for the number of "bad" pairs of polynomials. (Note that the total number is about $N^{n+m}$ since you assume the polynomials to be monic, and not $N^{n+m+2}$.)