Math Problem Statement
Answer,并且给出你解析的latex代码
Solution
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Transcribed Content
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Forms of Quadratic Integers
a. Let be the 3 positive prime numbers (mod 4). The quadratic integers in are precisely the ones of the form or where and are rational integers and and are either both even or both odd.
b. Let be the first 3 negative prime numbers (mod 4). The quadratic integers in are precisely the ones of the form where and are rational integers.
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:
- What are quadratic integers?
- How do we classify prime numbers modulo 4?
- What is the significance of in number theory?
- How does the parity (even or odd nature) of integers and affect the form of quadratic integers?
- Can you give examples of quadratic integers for specific values of ?
- Why are rational integers important in the context of quadratic integers?
- What other forms can quadratic integers take for different values of ?
- How are negative prime numbers treated differently in the context of quadratic integers?
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Math Problem Analysis
Mathematical Concepts
Quadratic Integers
Number Theory
Prime Numbers Modulo 4
Formulas
\( \frac{a + b\sqrt{d}}{2} \)
Theorems
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Suitable Grade Level
Advanced Mathematics
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