Math Problem Statement

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Solution

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  1. Forms of Quadratic Integers

    a. Let pp be the 3 positive prime numbers p2,3p \equiv 2, 3 (mod 4). The quadratic integers in Q(d)\mathbb{Q}(\sqrt{d}) are precisely the ones of the form a+bd2where a,b are rational integers.\frac{a + b\sqrt{d}}{2} \quad \text{where } a, b \text{ are rational integers.} d=1d = 1 or d=1d = -1 where aa and bb are rational integers and aa and bb are either both even or both odd.

    b. Let pp be the first 3 negative prime numbers d2d \equiv 2 (mod 4). The quadratic integers in Q(d)\mathbb{Q}(\sqrt{d}) are precisely the ones of the form a+bd2\frac{a + b\sqrt{d}}{2} where aa and bb 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:

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