I already posted this question at MSE [here](https://math.stackexchange.com/questions/586058/random-algebraic-numbers-are-linearly-disjoint-almost-surely), 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?



Random algebraic numbers are linearly disjoint almost surely?
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](http://www.ams.org/books/pspum/024/)) -- 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 $N^{n+m+2}$.)


