Ring of integers of a number field

Minkowski’s bound

Let O𝐾 be the ring of integers of a number field 𝐾. Then any ideal 𝔞 ⊴O𝐾 contains a nonzero element 𝛼 ∈𝔞 such that

|N⁡(𝛼)|≤𝑀𝐾N⁡(𝔞)

where Minkowski’s bound 𝑀𝐾 depends only on 𝐾 and is given by

𝑀𝐾=√|Δ𝐾:ℚ|(4𝜋)𝑟2𝑛!𝑛𝑛

where

• Δ𝐾:ℚ is the discriminant;

• 𝑛 =𝑟1 +2𝑟2 is the degree of 𝐾 :ℚ, 𝑟1 and (𝑟1,𝑟2) is the signature of 𝐾.¹

Proof sketch

proof

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

𝑆={(𝑎1,…,𝑎𝑟1,𝑥1,𝑦1,…,𝑥𝑟2,𝑦𝑟2:𝑟1∑𝑖=1|𝑎𝑖|+2𝑟2∑𝑖=1√𝑥2𝑖+𝑦2𝑖≤𝑛}

which looks like

in the complex case.

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Footnotes

1. 2022. Algebraic number theory course notes, ¶¶3.9–3.12, pp. 63–64 ↩