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Answer,并且给出你解析的latex代码

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Here is the content of the image transcribed and the corresponding LaTeX code:

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### Transcribed Content

1. Forms of Quadratic Integers

   a. Let \( p \) be the 3 positive prime numbers \( p \equiv 2, 3 \) (mod 4). The quadratic integers in \( \mathbb{Q}(\sqrt{d}) \) are precisely the ones of the form
   \[
   \frac{a + b\sqrt{d}}{2} \quad \text{where } a, b \text{ are rational integers.}
   \]
   \( d = 1 \) or \( d = -1 \) where \( a \) and \( b \) are rational integers and \( a \) and \( b \) are either both even or both odd.

   b. Let \( p \) be the first 3 negative prime numbers \( d \equiv 2 \) (mod 4). The quadratic integers in \( \mathbb{Q}(\sqrt{d}) \) are precisely the ones of the form \( \frac{a + b\sqrt{d}}{2} \) where \( a \) and \( b \) are rational integers.

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### LaTeX Code

```latex
\documentclass{article}
\usepackage{amsmath}
\begin{document}

\section*{1. Forms of Quadratic Integers}

\begin{enumerate}
    \item[a.] Let \( p \) be the 3 positive prime numbers \( p \equiv 2, 3 \pmod{4} \). The quadratic integers in \( \mathbb{Q}(\sqrt{d}) \) are precisely the ones of the form
    \[
    \frac{a + b\sqrt{d}}{2} \quad \text{where } a, b \text{ are rational integers.}
    \]
    \( d = 1 \) or \( d = -1 \) where \( a \) and \( b \) are rational integers and \( a \) and \( b \) are either both even or both odd.

    \item[b.] Let \( p \) be the first 3 negative prime numbers \( d \equiv 2 \pmod{4} \). The quadratic integers in \( \mathbb{Q}(\sqrt{d}) \) are precisely the ones of the form \( \frac{a + b\sqrt{d}}{2} \) where \( a \) and \( b \) are rational integers.
\end{enumerate}

\end{document}
```

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Here are some related questions to expand your understanding:

1. What are quadratic integers?
2. How do we classify prime numbers modulo 4?
3. What is the significance of \( \mathbb{Q}(\sqrt{d}) \) in number theory?
4. How does the parity (even or odd nature) of integers \( a \) and \( b \) affect the form of quadratic integers?
5. Can you give examples of quadratic integers for specific values of \( d \)?
6. Why are rational integers important in the context of quadratic integers?
7. What other forms can quadratic integers take for different values of \( d \)?
8. How are negative prime numbers treated differently in the context of quadratic integers?

**Tip:** When working with LaTeX, always compile your document incrementally to catch and correct errors early.

Explore the concept of quadratic integers in number theory, focusing on forms defined by prime numbers modulo 4. This page includes detailed transcriptions and LaTeX codes for different scenarios.

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