A unit is an element in a ring that has a multiplicative inverse. If is an algebraic integer which divides every algebraic integer in the field, is called a unit in that field. A given field may contain an infinity of units.
The units of are the elements relatively prime to . The units in which are squares are called quadratic residues.
All real quadratic fields have the two units .
The numbers of units in the imaginary quadratic field for , 2, ... are 4, 2, 6, 4, 2, 2, 2, 2, 4, 2, 2, 6, 2, ... (OEIS A092205). There are four units for , 4, 9, 16, ... (OEIS A000290; the square numbers), six units for , 12, 27, 48, ... (OEIS A033428; three times the square numbers), and two units for all other imaginary quadratic fields, i.e., , 5, 6, 7, 8, 10, 11, ... (OEIS A092206). The following table gives the units for small . In this table, is a cube root of unity.
units of | |
1 | , |
2 | |
3 | , , |