Imaginary quadratic field

An imaginary quadratic field $𝐾 = ℚ (\sqrt 𝑑)$ is a quadratic field with $𝑑 < 0$ , alg and hence signature $(𝑟 1, 𝑟 2) = (0, 1)$ .

Properties

The group of units $O \times 𝐾$ is ${1, - 1}$ except for $𝑑 = - 1$ , giving ring of integers Gaußian integers, or $𝑑 = - 3$ , giving Rationals adjoin sqrt(-3).

Proof of 1.

First consider the monogenic case, i.e. $𝑑 ≢ 4 1$ and hence $O 𝐾 = ℤ [\sqrt 𝑑]$ . Since the field norm of $𝛼 = 𝑎 + 𝑏 \sqrt 𝑑$
$N 𝐾 : ℚ (𝛼) = 𝑎 2 - 𝑏 2 𝑑$
where both terms are positive, the only ways to get $N 𝐾 : ℚ (𝛼) = 1$ are if

$𝑎 = \pm 1$ and $𝑏 = 0$ ; or

$𝑎 = 0$ , $𝑏 = \pm 1$ , and $𝑑 = - 1$ .

This exceptional case is Gaußian integers.

For $𝑑 \equiv 4 1$ , we have $O 𝐾 = ℤ [1 + \sqrt 𝑑 2]$ . The field norm of a generic $𝛼 = 𝑎 + 𝑏 1 + \sqrt 𝑑 2$ we have
$N 𝐾 : ℚ (𝛼) = (𝑎 + 𝑏 2) 2 - 𝑏 2 𝑑 4$
where both terms are positive, the only ways to get $N 𝐾 : ℚ (𝛼) = 1$ are if

$𝑎 = \pm 1$ and $𝑏 = 0$ ;

$𝑎 = 0$ , $𝑏 = \pm 1$ , and $𝑑 = - 3$ ; or

$𝑎 = \pm 1$ , $𝑏 = \mp 1$ , and $𝑑 = - 3$ .

This exceptional case is Rationals adjoin sqrt(-3).

Examples

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Imaginary quadratic field

Properties

Examples

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