Northcott theorem

There are only finitely many algebraic numbers of bounded degree and bounded Absolute multiplicative Weil height. The finite set can be enumerated effectively from the bounded coefficients of their primitive minimal polynomials.

The degree and Absolute multiplicative Weil height bounds are both essential.

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.