# 域中分數場

## 等價設定

${\displaystyle {\textbf {D}}}$ 係一個域。將${\displaystyle P={\textbf {D}}\backslash ({\textbf {D}}^{\times }\cup \{0\})=\{(a,b):a,b\in {\textbf {D}},b\neq 0\}}$

• 自反性${\displaystyle (a,b)\equiv (a,b)\iff ab=ab}$
• 對稱性${\displaystyle (a,b)\equiv (c,d)\iff ad=cb\iff (c,d)\equiv (a,b)}$
• 傳遞性${\displaystyle (a,b)\equiv (c,d)\iff ad=cb}$ 同時${\displaystyle (c,d)\equiv (e,f)\iff cf=de}$ ${\displaystyle af=be\iff (a,b)\equiv (e,f)}$

${\displaystyle P/\equiv }$ 叫做${\displaystyle Q({\textbf {D}})}$ ，而${\displaystyle Q({\textbf {D}})}$ 就係分數場（Field of Fractions）

## 加法同乘法

${\displaystyle Q({\textbf {D}})}$ 入面，都會有加法同乘法，而佢哋都係普通嘅加同乘。

${\displaystyle {\frac {a}{b}}={\frac {a'}{b'}}}$ ${\displaystyle {\frac {c}{d}}={\frac {c'}{d'}}}$

${\displaystyle ab'=a'b}$ ${\displaystyle cd'=c'd}$

${\displaystyle (ad+bc)b'd'=ab'd'+bb'cd'=a'bdd'+bb'c'd=(a'd'+b'c')bd}$

${\displaystyle {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {a'}{b'}}\cdot {\frac {c'}{d'}}}$
The text passage is a mathematical exposition on the construction of a field of fractions from a domain, including definitions, properties, and operations.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 域中分數場

## 等價設定

${\displaystyle {\textbf {D}}}$ 係一個域。將${\displaystyle P={\textbf {D}}\backslash ({\textbf {D}}^{\times }\cup \{0\})=\{(a,b):a,b\in {\textbf {D}},b\neq 0\}}$

• 自反性${\displaystyle (a,b)\equiv (a,b)\iff ab=ab}$
• 對稱性${\displaystyle (a,b)\equiv (c,d)\iff ad=cb\iff (c,d)\equiv (a,b)}$
• 傳遞性${\displaystyle (a,b)\equiv (c,d)\iff ad=cb}$ 同時${\displaystyle (c,d)\equiv (e,f)\iff cf=de}$ ${\displaystyle af=be\iff (a,b)\equiv (e,f)}$

${\displaystyle P/\equiv }$ 叫做${\displaystyle Q({\textbf {D}})}$ ，而${\displaystyle Q({\textbf {D}})}$ 就係分數場（Field of Fractions）

## 加法同乘法

${\displaystyle Q({\textbf {D}})}$ 入面，都會有加法同乘法，而佢哋都係普通嘅加同乘。

${\displaystyle {\frac {a}{b}}={\frac {a'}{b'}}}$ ${\displaystyle {\frac {c}{d}}={\frac {c'}{d'}}}$

${\displaystyle ab'=a'b}$ ${\displaystyle cd'=c'd}$

${\displaystyle (ad+bc)b'd'=ab'd'+bb'cd'=a'bdd'+bb'c'd=(a'd'+b'c')bd}$

${\displaystyle {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {a'}{b'}}\cdot {\frac {c'}{d'}}}$
user
The text passage is a mathematical exposition on the construction of a field of fractions from a domain, including definitions, properties, and operations.

TYPE: NO_QUESTION