 TOPICS # Unit

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 , , Eisenstein Unit, Fundamental Unit, Imaginary Unit, Prime Unit, Quadratic Residue, Root of Unity

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## References

Sloane, N. J. A. Sequences A000290/M3356, A033428, A092205, and A092206 in "The On-Line Encyclopedia of Integer Sequences."

Unit

## Cite this as:

Weisstein, Eric W. "Unit." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Unit.html