Quadratic field

Imaginary quadratic field

An imaginary quadratic field is a quadratic field with , alg and hence signature .

Properties

1. The group of units is except for , giving ring of integers Gaußian integers, or , giving Rationals adjoin sqrt(-3).

Proof of 1.

First consider the monogenic case, i.e. and hence . Since the field norm of

where both terms are positive, the only ways to get are if

• and ; or
• , , and .

This exceptional case is Gaußian integers.

For , we have . The field norm of a generic we have

where both terms are positive, the only ways to get are if

• and ;
• , , and ; or
• , , and .

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

Examples

• Rationals adjoin sqrt(-3)
• Rationals adjoin sqrt(-14)
• Rationals adjoin sqrt(-21)

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