For context, I was reviewing some old notes of mine about polynomials and got stuck in a proof (which I can only suppose it was mine and without any revision).
The problem was, let
$$h(t) = t^n + a_{n-1}t^{n-1} + \cdots + a_1t + a_0 \in \mathbb Z[t],$$
with $a_0 \neq 0$ and such that any real root $\rho$ of $h(t)$ satisfies $|\rho| < 1$.
Show that if
$$|a_{n-2}| > 1 + |a_{n-1}| + |a_{n-3}| + \cdots + |a_0|,$$
then $h(t)$ is irreducible over $\mathbb Q$.
Now the solution provided was (apparently inspired in the proof of Perron's Criterion)
Let $f(t) = a_{n-2}t^{n-2}$ and
$g(t) = t^n + a_{n-1}t^{n-1} + a_{n-3}t^{n-3} + \cdots + a_0$ and let $D = \{z \in \mathbb C : |z| < 1\}$.
Thus, for $|\lambda| = 1$,
$$|g(\lambda)| \leq 1 + |a_{n-1}| + |a_{n-3}| + \cdots + |a_0| < |a_{n-2}| = |f(\lambda)|.$$
Hence, by Rouché's Theorem, $f(t)$ and $h(t) = f(t) + g(t)$ have the same number of roots in $D$.
Since $f(t)$ has the root $0$ with multiplicity $n-2$, then $h(t)$ must have $n-2$ roots in $D$ and so it must have two roots $\lambda_1, \lambda_2 \notin D$.
By the hypothesis about real roots, it follows that $\lambda_1, \lambda_2 \notin \mathbb R$.Suppose now that $h(t)$ is reducible.
Then $h(t) = b(t)c(t)$ where we can suppose that $b(t), c(t) \in \mathbb Z[t]$ are monic and have degree less than the degree of $h(t)$.
Let $\beta_1, \ldots, \beta_r$ be the roots of $b(t)$ and $\gamma_1, \ldots, \gamma_s$ be the roots of $c(t)$, with $r + s = n$.
Then $\lambda_1, \lambda_2 \in \{\beta_1, \ldots, \beta_r\}$ or $\lambda_1, \lambda_2 \in \{\gamma_1, \ldots, \gamma_s\}$, for if each polynomial has one non-real root, then their coefficients cannot all be integer.
This is the point at which I disagree, meaning that I think the argument doesn't apply, because although we have $\lambda_1, \lambda_2 \notin \mathbb R$ because $\lambda_1, \lambda_2 \notin D$, we don't(?) have any guarantee that the roots which lie inside $D$ are real.
The proof beyond the point above goes on smoothly and I think it's even easy to guess where it's leading to, but now I though that perhaps the argument could be somewhat salvaged with the following conjecture I came up with, but which I wasn't able to prove or disprove.
Conjecture.
Let $f(t) = t^n + a_{n-1}t^{n-1} + \cdots + a_1t + a_0 \in \mathbb Z[t]$, $a_0\not=0$, and let $\alpha_1, \ldots, \alpha_n$ be the roots of $f(t)$.
If for some $i_0$ we have $|\alpha_{i_0}| \geq 1$ and $\alpha_{i_0} \notin \mathbb R$, then there is some $i_1 \neq i_0$ with $|\alpha_{i_1}| \geq 1$.
I know that $a_{n-1}$ is the sum of the roots, $a_{n-2}$ the sum of the products of two roots, etc, until $a_0$ the product of all roots, but I wasn't able to use that information to prove the result.
Sure an independent proof of the original problem would also work, but I would prefer to use most of the argument.
Best Answer
The conjecture is true since $\overline{\alpha_{i_0}}$ is also a root for $f(t)$, a polynomial with real coefficients. Since $\alpha_{i_0}\notin\Bbb R$, we have $\overline{\alpha_{i_0}}\notin\Bbb R$ and $\overline{\alpha_{i_0}}\not=\alpha_{i_0}$. We also have $|\overline{\alpha_{i_0}}|=|\alpha_{i_0}|\ge1$.
As seen from the proof above, the conjecture is true for all polynomials with real coefficients.
Let us salvage the proof following the suggestion by the original poster. Continue the argument right after "let $\beta_1, \ldots, \beta_r$ be the roots of $b(t)$ and $\gamma_1, \ldots, \gamma_s$ be the roots of $c(t)$, with $r + s = n$." Since both $b(t)$ and $c(t)$ are monic polynomials in $\Bbb Z[t]$, neither can have exactly one non-real root outside $D$, thanks to the conjecture/theorem. Hence, $\lambda_1, \lambda_2 \in \{\beta_1, \ldots, \beta_r\}$ or $\lambda_1, \lambda_2 \in \{\gamma_1, \ldots, \gamma_s\}$. WLOG, suppose the former is true. Then all roots of $c(t)$ are inside $D$. So is their product. However, their product is either the constant term of $c(t)$ or the negation of it, which is an integer. This is a contradiction.