Minkowski's bound

Let $O 𝐾$ be the ring of integers of a number field $𝐾$ . Then any ideal $𝔞 ⊴ O 𝐾$ contains a nonzero element $𝛼 \in 𝔞$ such that

| N (𝛼) | \leq 𝑀 𝐾 N (𝔞)

where Minkowski's bound $𝑀 𝐾$ depends only on $𝐾$ and is given by

𝑀 𝐾 = \sqrt | Δ 𝐾 : ℚ | (4 𝜋) 𝑟 2 𝑛! 𝑛 𝑛

where

$Δ 𝐾 : ℚ$ is the discriminant;
$𝑛 = 𝑟 1 + 2 𝑟 2$ is the degree of $𝐾 : ℚ$ , $𝑟 1$ and $(𝑟 1, 𝑟 2)$ is the signature of $𝐾$ .¹

Proof sketch

#missing/proof

We apply Minkowski's convex body theorem to the convex symmetric compact region

𝑆 = {(𝑎 1, \dots, 𝑎 𝑟 1, 𝑥 1, 𝑦 1, \dots, 𝑥 𝑟 2, 𝑦 𝑟 2 : 𝑟 1 \sum 𝑖 = 1 | 𝑎 𝑖 | + 2 𝑟 2 \sum 𝑖 = 1 \sqrt 𝑥 2 𝑖 + 𝑦 2 𝑖 \leq 𝑛}

which looks like

in the complex case.

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Footnotes