Maximum absolute value of polynomial coefficients Suppose we have a polynomial in integer coefficients 
$$p = p_0 + p_1 x + p_2 x^2 + \ldots + p_n x^n, p_k \in \mathbb{Z}$$
Now define $M(p)$ as the maximum absolute value of the coefficients of $p$, i.e.
$$M(p) = \max \{|p_k| \: |\: 0 \leq k \leq n\}$$
Is it true that for any polynomial factor $q$ of $p$, $M(q) \leq M(p)$? I'm not sure whether this is obvious and I'm just missing something, or whether my train of thought is just completely off...

No. There is a famous example of a factor of $x^{105}-1$ 
\begin{align}
\Phi_{105}(x) = & \; x^{48} + x^{47} + x^{46} - x^{43} - x^{42} - 2 x^{41} - x^{40} - x^{39} + x^{36} + x^{35} + x^{34} \\
& {} + x^{33} + x^{32} + x^{31} - x^{28} - x^{26} - x^{24} - x^{22} - x^{20} + x^{17} + x^{16} + x^{15} \\
& {} + x^{14} + x^{13} + x^{12} - x^9 - x^8 - 2 x^7 - x^6 - x^5 + x^2 + x + 1
\end{align}
See cyclotomic polynomials
Also $x^4+1=(x^2+1)^2-2x^2$ has the factorisation $(x^2+\sqrt 2 x + 1)(x^2-\sqrt 2 x +1)$ which isn't integral, but does suggest that lower degree counterexamples may exist.
