Unit

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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.

See also

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."

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Unit

Cite this as:

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

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