= Minimal-polynomial coefficient proof of Northcott finiteness
{title2=$|a_j|\leq2^D B^D\quad\text{if }[\mathbb Q(\alpha):\mathbb Q]\leq D,\ H(\alpha)\leq B$}
For the primitive minimal polynomial $a_d\prod_j(T-\alpha_j)$, the <height-Mahler measure formula> gives $H(\alpha)^d=a_d\prod_j\max(1,|\alpha_j|)$. Expanding the product bounds every integer coefficient by $2^D B^D$. There are finitely many bounded coefficient lists and each has finitely many roots, proving finiteness of the algebraic numbers. On a projective coordinate chart, every coordinate ratio has height at most the point's <projective height>; applying the scalar bound proves the projective <Northcott theorem> too. A degree bound is essential, as the infinitely many <roots of unity> all have height one.
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