[Math] Elementary proof for Hilbert’s irreducibility theorem

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I have tried to find a complete proof for Hilbert's irreducibility theorem, but everything I found was way beyond my level of understanding.

I am only interested in the simple case where the polynomial is in two variables over the rationals. Specifically, if $f\in \mathbb{Q}[T,X]$ be an irreducible polynomial, then there exist infinitely many $t_j\in\mathbb{Q}$ such that $f(t_j,X)\in\mathbb{Q}[X]$ is irreducible.

Is there a way to prove this using mostly elementary results?

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Kaltofen's 1985 proof (Wayback machine) seems completely elementary and effective.

E. Kaltofen. Polynomial-time reductions from multivariate to bi- and univariate integral polynomial factorization. SIAM J. Comput., 14(2):469-489, 1985; DOI: 10.1137/0214035.

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Elementary Proof of the Infinitude of Completely Split Primes – Number Theory

By the primitive element theorem, $K=\mathbb{Q}(\alpha)$ for some nonzero $\alpha \in K$, and we may assume that the minimal polynomial $f(x)$ of $\alpha$ has integer coefficients. Let $\Delta$ be the discriminant of $f$. Since $K/\mathbb{Q}$ is Galois, a prime $p \nmid \Delta$ splits completely in $K$ if and only if there is a degree $1$ prime above $p$, which is if and only if $p | f(n)$ for some $n \in \mathbb{Z}$. Suppose that the set $P$ of such primes is finite. Enlarge $P$ to include the primes dividing $\Delta$. Let $t$ be a positive integer such that $\operatorname{ord}_p t> \operatorname{ord}_p f(0)$ for all $p \in P$. For any integer $m$, we have $f(mt) \equiv f(0) \;(\bmod \; t)$, so $\operatorname{ord}_p f(mt) = \operatorname{ord}_p f(0)$ for all $p \in P$. But $f(mt) \to \infty$ as $m \to \infty$, so eventually it must have a prime factor outside $P$, contradicting the definition of $P$.

[Math] Irreducibililty tests for cubic and quartic polynomials over finite fields

These results have been known for many years. Here are some references and further developments; full bibliographic details are at the end of this answer.

Reducibility criteria for cubics over finite fields of characteristic at least $5$ were given already in 1906 by Dickson. In fact he gave conditions for a cubic to have each of the possible factorization types. Reducibility criteria for cubics in characteristic $2$ were given in 1966 by Berlekamp, Rumsey and Solomon (see also their 1967 paper). Those criteria are slightly different than your Theorem A. Your Theorems A and B appear to have been published for the first time in 1975 by Williams, who also gave criteria for each of the possible factorization types.

Reducibility criteria for quartics over prime fields of odd order were given by Skolem in 1952. In 1969, Leonard gave a different proof of Skolem's result, which extends at once to non-prime fields of odd order. Finally, reducibility criteria for quartics in characteristic $2$ were given by Leonard and Williams in 1972.

Your Theorems A and B can be generalized in various ways. Bluher's 2004 paper generalizes your Theorem A to polynomials of the form $x^{p^k+1}+ax+b$. In a series of papers culminating in 1980, Agou determined all instances when $f(L(x))$ is irreducible over $\mathbf{F}_{p^s}$, where $f,L\in\mathbf{F}_{p^s}[x]$ and all terms of $L(x)$ have degree $p^i$ with $i\ge 0$. The case $f=x+c_3$ and $L=x^3+c_2 x$ with $p=3$ yields your Theorem B. Subsequently, a simpler proof of Agou's results was given by Cohen in 1982; following two related papers by Moreno, Cohen gave an even simpler proof in 1989.

References:

S. Agou, Irréducibilité des polynômes $f(\sum_{i=0}^m a_i X^{p^{ri}})$ sur un corps fini $\mathbf{F}_{p^s}$, Canadian Mathematical Bulletin 23 (1980), 207-212
E. R. Berlekamp, H. Rumsey and G. Solomon, Solutions of algebraic equations in field of characteristic 2, Jet Propulsion Lab. Space Programs Summary No. 4 (1966), 37-39
E. R. Berlekamp, H. Rumsey and G. Solomon, On the solution of algebraic equations over finite fields, Information and Control 10 (1967), 553-564.
A. W. Bluher, On $x^{q+1}+ax+b$, Finite Fields and their Applications 10 (2004), 285-305
S. D. Cohen, The irreducibility of compositions of linear polynomials over a finite field, Compositio Mathematica 47 (1982), 149-152
S. D. Cohen, The reducibility theorem for linearised polynomials over finite fields, Bulletin of the Australian Mathematical Society 40 (1989), 407-412
L. E. Dickson, Criteria for the irreducibility of functions in a finite field, Bulletin of the American Mathematical Society 13 (1906), 1-8
P. A. Leonard, On factoring quartics (mod $p$), Journal of Number Theory 1 (1969), 113-115
P. A. Leonard and K. S. Williams, Quartics over GF($2^n$), Proceedings of the American Mathematical Society 36 (1972), 347-350
Th. Skolem, The general congruence of 4th degree modulo $p$, $p$ prime, Nordisk matematisk tidskrift 34 (1952), 73-80
K. S. Williams, Note on cubics over GF($2^n$) and GF($3^n$), Journal of Number Theory 7 (1975), 361-365

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Related Solutions

Elementary Proof of the Infinitude of Completely Split Primes – Number Theory

[Math] Irreducibililty tests for cubic and quartic polynomials over finite fields

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