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I would be very grateful if you give me a hint on problem 9, section 3.6 of Hungerford Algebra, regarding factorization in polynomial rings, saying that:

Suppose $f(x)= a_0+a_1x+...+a_kx^k+...+a_nx^n$ is a polynomial of degree $n$, such that for some prime $p$ and for some $k$ with $0\lt k\lt n$ we have that:
$1)\ \ p$ divides $a_0$ and ... and $a_{k-1}$
$2)\ \ p$ does not divide $a_k$.
$3)\ \ p^2$ does not divide $a_0$.
$4) \ \ p$ does not divide $a_n$.
Prove that $f$ has an irreducible factor of degree at least $k$ over $\mathbb Z$.

I think the idea is to incorporate Eisenstein's criterion, as the first three assumptions imply that $a_0+a_1x^1+...+a_kx^k$ is irreducible, but have no idea where to go from there. Thanks for your time in advance!

Bill Dubuque
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Sean
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2 Answers2

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Ok here is your hint. Look at the irreducible factors of $f$. Exactly one will have a constant term divisible by $p$. Show this factor has degree $\geq k$, by looking at its highest term whose coefficient is divisible by $p$.

Rene Schipperus
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Hint $\ $ Consider any factorization $\,f = gh\,$ modulo $\,p.\,$ The hypotheses imply precisely one of $\,g,h,\,$ say $\,g,\,$ have constant term $\,g_0\!\not\equiv 0,\,$ and $\, h\equiv h_j x^j +\cdots,\,\ h_j\not\equiv 0.\ $ Therefore

$\qquad\quad gh\equiv f\ \Rightarrow\ (g_0\! + \cdots)(h_j x^j + \cdots)\, \equiv\, a_k x^k + \cdots\,\Rightarrow\,j\ge k,\,\ $ so $\,\ \deg h\ge k$

Remark $\ $ Essentially we are viewing the polynomials as (formal) power series modulo $\,p,\,$ and comparing their orders. The algebra behind this will become much clearer when one studies valuation theory, esp. the theory of Newton Polygons/Polyhedra, which significantly generalize Eisenstein and related irreducibility criteria. See this answer for links to some excellent expositions.

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Bill Dubuque
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