I'm trying to understand a proof of the following theorem (from section II of Hall's paper [*An Isomorphism Between Linear Recurring Sequences and Algebraic Rings*](http://www.ams.org/journals/tran/1938-044-02/S0002-9947-1938-1501967-8/S0002-9947-1938-1501967-8.pdf)):



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> If $F(a\_1, \ldots, a\_k)$ is a polynomial in $a\_1, \ldots, a\_k$ with integer coefficients and $F = 0$ whenever $a\_1, \ldots, a\_k$ are integers such that $f(x) = x^k - a\_1x^{k - 1} - \cdots - a\_k$ is irreducible, then $F \equiv 0$. 
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The proof provided was that taking $f(x)$ modulo $p$ for some $p$ (prime?) irreducible mod $q$, we find that $F = 0$ (modulo $p$). Since $F \equiv 0$ (mod $p$) for any appropriate $a\_1, \ldots, a\_k$ for arbitrary $p$, we have that $F \equiv 0$.


I'm confused about the first part of the proof because irreducibility over integers doesn't imply irreducibility modulo $p$ for any prime $p$. Also, does it make sense to consider $q$ to be an arbitrary positive integer?


Does the theorem assume that we're considering irreducibility mod $q$? Does this change anything? Also, what does "e" on p. 200 (after Theorem 4.1) refers to? There was no mention of it earlier in the paper and I don't see why it should be the constant $e$.


Note: This is cross-posted from math.SE (and slightly edited): <https://math.stackexchange.com/questions/459173/if-fa-1-ldots-a-k-0-whenever-a-1-ldots-a-k-are-integers-such-that-fx>



If $F(a_1,\ldots,a_k)=0$ whenever $a_1,\ldots,a_k$ are integers such that $f(x)=x^k-a_1x^{k-1}-\cdots-a_k$ is irreducible, then $F\equiv0$
Pick two distinct primes $p$ and $q$. Now choose any monic irreducible polynomial $f\_q$ of degree $k$ modulo $q$ (i.e. in ${\bf F}\_q$); these exist. Also choose $a\_1, \dots, a\_k$ arbitrary mod $p$ and let $f\_p = x^k - a\_1 x^{k-1} + \dots + (-1)^k a\_k$. Then by CRT, there is a common lift $f(x) \in {\bf Z}[x]$ which agrees with $f\_p$ mod $p$ and with $f\_q$ mod $q$. Since $f\_q$ is irred, so is $f$. So $F(a\_1, \dots, a\_k) = 0$ for every such choice, and so $F \equiv 0 \pmod p$ etc. 


Aso for your second question, I'm not sure, but it looks like you pick $h$ to be one of the $h\_i$ from equation (4.3) and then $e$ is the associated $e\_i$.


