Logarithmic 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 Logarithmic embedding $𝐿 : 𝐾 \times \to ℝ 𝑟 1 + 𝑟 2$ is a group homomorphism defined by #m/def/num/alg

𝐿 (𝛼) = (l n | 𝜎 1 (𝛼) |, \dots, l n | 𝜎 𝑟 1 (𝛼) |, l n | 𝜏 1 (𝛼) | 2, \dots, l n | 𝜏 𝑟 2 | 2) .

We call $𝐺 = 𝐿 (O * 𝐾)$ the unit lattice for $𝐾$ , and its covolume is called the regulator.

Properties

The norm of an element is related to the sum of its image by

l n | N (𝛼) | = Σ 𝐿 (𝛼)

where $Σ : ℝ 𝑟 1 + 𝑟 2 \to ℝ$ is the summation map. 2. $k e r (𝐿 ↾ O 𝐾) = 𝑊 𝐾$ , the group of roots of unity, by Kronecker's root of unity lemma.

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