A unit is an element in a ring that has a multiplicative inverse. If a is an algebraic integer which divides every algebraic integer in the field, a is called a unit in that field. A given field may contain an infinity of units.

The units of Z_n are the elements relatively prime to n. The units in Z_n which are squares are called quadratic residues.

All real quadratic fields Q(sqrt(D)) have the two units +/-1.

The numbers of units in the imaginary quadratic field Q(sqrt(-D)) for D=1, 2, ... are 4, 2, 6, 4, 2, 2, 2, 2, 4, 2, 2, 6, 2, ... (OEIS A092205). There are four units for D=1, 4, 9, 16, ... (OEIS A000290; the square numbers), six units for D=3, 12, 27, 48, ... (OEIS A033428; three times the square numbers), and two units for all other imaginary quadratic fields, i.e., D=2, 5, 6, 7, 8, 10, 11, ... (OEIS A092206). The following table gives the units for small D. In this table, omega is a cube root of unity.

Dunits of Q(sqrt(-D))
1+/-1, +/-i
3+/-1, +/-omega, +/-omega^2

See also

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

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

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Cite this as:

Weisstein, Eric W. "Unit." From MathWorld--A Wolfram Web Resource.

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