Hilbert's Irreduicibility Theorem

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Hilbert's Irreduicibility Theorem : 

Let $\Q$ be the field of rational numbers and $f_1(x_1, x_2, ..., x_n, y_1, y_2, ..., y_m), f_1(x_1, x_2, ..., x_n, y_1, y_2, ..., y_m),...,f_r(x_1, x_2, ..., x_n, y_1, y_2, ..., y_m)$ be a set of irreducible polynomials in $\Q(x_1, x_2, ..., x_n)[y_1, y_2, ..., y_m]$ . Then Hilbert's Irreduicibility Theorem tells that there exists an n-tuple $(a_1, a_2, ..., a_n)$ in $\Q^n$ such that $f_1(a_1, a_2, ..., a_n, y_1, y_2, ..., y_m), f_2(a_1, a_2, ..., a_n, y_1, y_2, ..., y_m),..., f_r (a_1, a_2, ..., a_n, y_1, y_2, ..., y_m)$ are irreducible in $\Q[ y_1, y_2, ..., y_m]$.

A cute application of above theorem is that, If a polynomial $p(x)$ in $\Q[x]$ takes square value at every points in $\Q$ then $p(x)$ is itself a square in $\Q[x]$. (Hint : Apply Hilbert's irreducibilty to f(x, y) = y^2-g(x).)

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