Quadratic field

Imaginary quadratic field

An imaginary quadratic field is a quadratic field with , #m/def/num/alg and hence signature .

Properties

  1. The group of units is except for , giving ring of integers , or , giving .
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 .

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 .

Examples

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