= Height function
{wiki}
= Arithmetic height
{synonym}
A <height function> measures arithmetic complexity rather than physical size. For a reduced <rational number> $a/b$, its multiplicative height is $H(a/b)=\max(|a|,|b|)$, and its logarithmic height is $h(a/b)=\log H(a/b)$. Thus $1001/1000$ is close to one in real size but has height $1001$.
For an <algebraic number> $\alpha$ of degree $d$, let its primitive irreducible <integer> <polynomial> have leading coefficient $a_d$ and complex roots $\alpha_1,\ldots,\alpha_d$. The <Absolute logarithmic Weil height> is $d^{-1}(\log|a_d|+\sum_i\log\max(1,|\alpha_i|))$. Bounding both height and degree leaves finitely many possibilities, making heights useful in <Diophantine equations> and finiteness arguments.
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