Word Count : 421

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To show that there are infinitely many values of \(\alpha\) for which the polynomial \[ f(x) = x^7 + 15x^2 - 30x + \alpha \] is irreducible over \(\mathbb{Q}[x]\), we will use Eisenstein's criterion and modular reduction arguments. --- ### Step 1: Check Eisenstein's Criterion Eisenstein's criterion states that if there exists a prime \( p \) such that: - \( p \) divides all coefficients except the leading coefficient, - \( p^2 \) does not divide the constant term, then the polynomial is irreducible in \(\mathbb{Q}[x]\). Rewriting the polynomial: \[ f(x) = x^7 + 15x^2 - 30x + \alpha. \] - The leading coefficient ____ ___ _________ ___ _____ _____ _______ ______ ____ ______ __________.
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