# 有序體 (ordered field)

[代數](..) ⟩ [體]() ⟩ 有序體 (ordered field)

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A [**field**]() $$\mathbb{F}$$ is called <mark style="color:purple;">**ordered**</mark> if there is a <mark style="color:yellow;">**subset**</mark> $$\mathbb{P}$$ of $$\mathbb{F}$$, called the <mark style="color:yellow;">**set of positive elements**</mark> 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.  $$\forall x \in \mathbb{F}$$, <mark style="color:orange;">**one and only one**</mark> of the following three is true:

    (1) $$x=0$$ (2) $$x \in P$$ (3) $$-x \in P$$
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A [ﬁeld]() F is called an <mark style="color:purple;">**ordered ﬁeld**</mark> if F is endowed with a [quan-xu-total-ordering](../../set/binary/quan-xu-total-ordering "mention") <mark style="color:yellow;">**≤**</mark> that satisﬁes：

* (<mark style="color:yellow;">**Ｏ4**</mark>)：x <mark style="color:yellow;">**≤**</mark> y => x + z <mark style="color:yellow;">**≤**</mark> y + z
* (<mark style="color:yellow;">**Ｏ5**</mark>)：x ≥ 0, y ≥ 0 => xy ≥ 0

:point\_right: Understanding Analysis
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{% tab title="⭐️ 重點" %}
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[複數](../../num/complex)系<mark style="color:red;">**不是**</mark>有序體， 🎖 證明： :point\_right: <img src="https://487293287-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-M5-JmwIjjSzWft9bo8C%2F-M5-Jox2a6qC-zOAFROX%2F-M5-JpIWYpZOT9IWCXUD%2F811FA237-BDFC-44EC-BF0F-AEB950C99D22.jpeg?generation=1586995877874052&#x26;alt=media" alt="" data-size="line">
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{% tab title="👥 相關" %}
* 「[real](../../num/real "mention")」是一個<mark style="color:purple;">**有序體**</mark>。
* what is an [quan-xu-total-ordering](../../set/binary/quan-xu-total-ordering "mention")?
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{% tab title="📗 參考" %}
* Understanding Analysis ⟩ 8.6 A Construction of R From Q, Def. 8.6.5 (p.299)
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