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prime ideal decomposition in quadratic extensions of $\mathbb{Q}$ (Theorem)

Let $ K$ be a quadratic number field, i.e. $ K=\mathbb{Q}(\sqrt{d})$ for some square-free integer $ d$. The discriminant of the extension is

$\displaystyle D_{K/\mathbb{Q}}=\begin{cases}d, & \text{ if } d\equiv 1 \ \operatorname{mod}\ 4,\\ 4d, & \text{ if } d\equiv 2,3 \operatorname{mod}\ 4.\end{cases}$    

Let $ \mathcal{O}_K$ denote the ring of integers of $ K$. We have:
$\displaystyle \mathcal{O}_K\cong \begin{cases}\mathbb{Z}\oplus \frac{1+\sqrt{d}... ...sqrt{d}\mathbb{Z}, & \text{ if } d\equiv 2,3 \operatorname{mod}\ 4. \end{cases}$    

Prime ideals of $ \mathbb{Z}$ decompose as follows in $ \mathcal{O}_K$:
Theorem 1   Let $ p\in \mathbb{Z}$ be a prime.
  1. If $ p\mid d$ (divides), then $ p\mathcal{O}_K=(p,\sqrt{d})^2$;
  2. If $ d$ is odd, then
    $\displaystyle 2\mathcal{O}_K=\begin{cases}(2,1+\sqrt{d})^2, & \text{ if } d\equ... ... 8,\\ \text{prime}, & \text{ if } d\equiv 5\ \operatorname{mod}\ 8. \end{cases}$    

  3. If $ p\neq 2$, $ p$ does not divide $ d$, then
    $\displaystyle p\mathcal{O}_K=\begin{cases}(p,n+\sqrt{d})(p,n-\sqrt{d}), & \text... ...t{prime}, & \text{ if $d$\ is not a square } \operatorname{mod}\ p. \end{cases}$    

Bibliography

1
Daniel A.Marcus, Number Fields. Springer, New York.



"prime ideal decomposition in quadratic extensions of $\mathbb{Q}$" is owned by alozano.
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See Also: calculating the splitting of primes, examples of prime ideal decomposition in number fields, prime ideal decomposition in cyclotomic extensions of $\mathbb{Q}$, number field, splitting and ramification in number fields and Galois extensions

Keywords:  quadratic field, prime decomposition, splitting

Attachments:
proof of prime ideal decomposition in quadratic extensions of $\mathbb{Q}$ (Proof) by Wkbj79
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Cross-references: odd, divides, prime, prime ideals, ring of integers, extension, discriminant, integer, square-free, quadratic number field
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This is version 4 of prime ideal decomposition in quadratic extensions of $\mathbb{Q}$, born on 2003-08-22, modified 2006-07-19.
Object id is 4643, canonical name is PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ.
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Classification:
AMS MSC: 11R11 (Number theory :: Algebraic number theory: global fields :: Quadratic extensions)
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