Signature of a number field

Let $𝐾 = ℚ (𝜗)$ be a number field of degree $𝑛$ . Then $𝐾$ has $𝑛$ distinct field embeddings $𝜎 𝑖 : 𝐾 ↪ ℂ$ into $ℂ$ , which are either real or unreal. Moreover, the unreal embeddings come in conjugate pairs. We label the embeddings such that $𝜎 1, \dots, 𝜎 𝑟 1$ are the real embeddings and $𝜏 1, \dots, 𝜏 𝑟 2$ are representatives of conjugate pairs of unreal embeddings. Then

𝑛 = 𝑟 1 + 2 𝑟 2

and $(𝑟 1, 𝑟 2)$ is called the signature of $𝐾$ .

Sage

Given a number field K, we can get the signature using K.signature().

Signature of a number field

See also