🔰有序體 (ordered field)

代數 ⟩ 體 ⟩ 有序體 (ordered field)

A field $\mathbb{F}$ is called ordered if there is a subset $\mathbb{P}$ of $\mathbb{F}$, called the set of positive elements of $\mathbb{F}$, satisfying the following:

1. $\forall x, y \in P, x + y \in P$

2. $\forall x, y \in P, x \cdot y \in P$

3. $\forall x \in \mathbb{F}$, one and only one of the following three is true:

   (1) $x=0$ (2) $x \in P$ (3) $-x \in P$

A field F is called an ordered field if F is endowed with a 全序╱total ordering ≤ that satisfies：

• (Ｏ4)：x ≤ y => x + z ≤ y + z

• (Ｏ5)：x ≥ 0, y ≥ 0 => xy ≥ 0

👉 Understanding Analysis

⭐️ 重點

複數系不是有序體， 🎖 證明： 👉

👥 相關

• 「實數 ℝ」是一個有序體。

• what is an 全序╱total ordering?

📗 參考

• Understanding Analysis ⟩ 8.6 A Construction of R From Q, Def. 8.6.5 (p.299)