Minkowski embedding

Let $𝐾$ be a number field with signature $(𝑟 1, 𝑟 2)$ with real embeddings ${𝜎 𝑖} 𝑟 1 𝑖 = 1$ and representative unreal embeddings ${𝜏 𝑖} 𝑟 2 𝑖 = 1$ . The Minkowski embedding #m/def/num/alg

𝜄 : 𝐾 ↪ ℝ 𝑟 1 \times ℝ 2 𝑟 2 ≅ ℝ 𝑛

is determined by $(𝜎 𝑚, \dots, 𝜎 𝑟 1, 𝜏 1, \dots, 𝜏 𝑟 2)$ where we identify $ℂ 𝑟 2 ≅ ℝ 2 𝑟 2$ .

Fundamental property

Let $O 𝐾$ be the ring of integers. Then $𝜄 (O 𝐾)$ is a Classical lattice of rank $𝑛$ , moreover it has covolume #m/thm/num/alg

c o v o l 𝜄 (O 𝐾) = 2 - 𝑟 2 \sqrt | Δ 𝐾 : ℚ |

where $Δ 𝐾 : ℚ$ is the discriminant.¹

Proof

Suppose ${𝛼 𝑖} 𝑛 𝑖 = 1 \subset O 𝐾$ is an Integral basis for $𝐾$ . It suffices to show that ${𝜄 (𝛼 𝑖)} 𝑛 𝑖 = 1$ form a basis for $ℝ 𝑛$ . To this end, let

𝐴 1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝜎 1 (𝛼 1) \dots 𝜎 1 (𝛼 𝑛) ⋮ ⋱ ⋮ 𝜎 𝑟 1 (𝛼 1) \dots 𝜎 𝑟 1 (𝛼 𝑛) ℜ 𝜏 1 (𝛼 1) \dots ℜ 𝜏 1 (𝛼 𝑛) ℑ 𝜏 1 (𝛼 1) \dots ℑ 𝜏 1 (𝛼 𝑛) ⋮ ⋱ ⋮ ℜ 𝜏 𝑟 2 (𝛼 1) \dots ℜ 𝜏 𝑟 2 (𝛼 𝑛) ℑ 𝜏 𝑟 2 (𝛼 1) \dots ℑ 𝜏 𝑟 2 (𝛼 𝑛) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ \in M 𝑛, 𝑛 (ℂ)

be the matrix containing all these embeddings of the $𝛼 𝑖$ . We now apply the following elementary row operations:

Add $𝑖 ℑ 𝜏 𝑗 (𝛼 𝑘)$ to $ℜ 𝜏 𝑗 (𝛼 𝑘)$ giving

𝐴 2 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝜎 1 (𝛼 1) \dots 𝜎 1 (𝛼 𝑛) ⋮ ⋱ ⋮ 𝜎 𝑟 1 (𝛼 1) \dots 𝜎 𝑟 1 (𝛼 𝑛) 𝜏 1 (𝛼 1) \dots 𝜏 1 (𝛼 𝑛) ℑ 𝜏 1 (𝛼 1) \dots ℑ 𝜏 1 (𝛼 𝑛) ⋮ ⋱ ⋮ 𝜏 𝑟 2 (𝛼 1) \dots 𝜏 𝑟 2 (𝛼 𝑛) ℑ 𝜏 𝑟 2 (𝛼 1) \dots ℑ 𝜏 𝑟 2 (𝛼 𝑛) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .

Multiply $ℑ 𝜏 𝑗 (𝛼 𝑘)$ by $- 2 𝑖$ giving

𝐴 3 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝜎 1 (𝛼 1) \dots 𝜎 1 (𝛼 𝑛) ⋮ ⋱ ⋮ 𝜎 𝑟 1 (𝛼 1) \dots 𝜎 𝑟 1 (𝛼 𝑛) 𝜏 1 (𝛼 1) \dots 𝜏 1 (𝛼 𝑛) - 2 𝑖 ℑ 𝜏 1 (𝛼 1) \dots - 2 𝑖 ℑ 𝜏 1 (𝛼 𝑛) ⋮ ⋱ ⋮ 𝜏 𝑟 2 (𝛼 1) \dots 𝜏 𝑟 2 (𝛼 𝑛) - 2 𝑖 ℑ 𝜏 𝑟 2 (𝛼 1) \dots - 2 𝑖 ℑ 𝜏 𝑟 2 (𝛼 𝑛) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .

Add $𝜏 𝑗 (𝛼 𝑘)$ to $- 2 𝑖 ℑ 𝜏 𝑗 (𝛼 𝑘)$ giving

𝐴 4 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝜎 1 (𝛼 1) \dots 𝜎 1 (𝛼 𝑛) ⋮ ⋱ ⋮ 𝜎 𝑟 1 (𝛼 1) \dots 𝜎 𝑟 1 (𝛼 𝑛) 𝜏 1 (𝛼 1) \dots 𝜏 1 (𝛼 𝑛) ―― 𝜏 1 (𝛼 1) \dots ―― 𝜏 1 (𝛼 𝑛) ⋮ ⋱ ⋮ 𝜏 𝑟 2 (𝛼 1) \dots 𝜏 𝑟 2 (𝛼 𝑛) ――― 𝜏 𝑟 2 (𝛼 1) \dots ――― 𝜏 𝑟 2 (𝛼 𝑛) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .

We see now that $𝐴 4 = 𝑇 (𝛼 1, \dots, 𝛼 𝑛)$ as defined in Discriminant of a separable extension, and thus

c o v o l 𝜄 (O K) = | d e t 𝐴 1 | = 2 - 𝑟 2 | d e t 𝑇 (𝛼 1, \dots, 𝛼 𝑛) | = 2 - 𝑟 2 \sqrt | Δ 𝐾 : ℚ | \neq 0

as required.

Norm

This generalizes by ^P1 for an ideal $𝔞 ⊴ O 𝐾$ so that

c o v o l 𝜄 (O 𝐾) = 2 - 𝑟 2 \sqrt | Δ 𝐾 : ℚ | N (𝔞)

whence we define the norm on $ℝ 𝑟 1 \times ℝ 2 𝑟 2$ by

N (𝑎 1, \dots, 𝑎 𝑟 1, 𝑥 1, 𝑦 1, \dots, 𝑥 𝑟 2, 𝑦 𝑟 2) = 𝑎 1 \dots 𝑎 𝑟 1 (𝑥 21 + 𝑦 21) \dots (𝑥 2 𝑟 2 + 𝑦 2 𝑟 2)

so that

N (𝜄 (𝛼)) = N 𝐾 : ℚ (𝛼) .

Properties

Minkowski's bound

#state/tidy | #lang/en | #SemBr

2022. Algebraic number theory course notes, ¶3.1, p. 58 ↩

Minkowski embedding

Fundamental property

Norm

Properties

Footnotes