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Units

Units are elements of a set having a multiplicative inverse belonging to that set. (The set of units form a multiplicative group.) Algebraic integer units have complex norm 1 and are thus complex roots of unity.

Imaginary quadratic integer rings have only a few units (see Theorem I1). In the table below, $\omega = \frac{-1}{2} + \frac{\sqrt{-3}}{2}$, one of the complex cubic roots of 1.

 $d \leq -5$ –1, 1 d = − 3 ω, − ω,ω2, − ω2,1, − 1 d = − 2 –1, 1 d = − 1 − 1,1, − i,i

Real quadratic fields have infinitely many units (see Theorem R1), which are all powers of each other, some multiplied by –1. For that reason, the table below can't be complete like the table above.

An inefficient way to find a unit of a real field $\mathbb{Z}[\sqrt{d}]$ other than 1 or –1 is to try each positive value of b starting with 1 and going up until $\sqrt{1 + db^2}$ is an integer.

 d 2 $1 + \sqrt{2}$ 3 $2 + \sqrt{3}$ 4 N/A 5 $2 + \sqrt{5}$ 6 $5 + 2 \sqrt{6}$ 7 $8 + 3 \sqrt{7}$ 8 N/A, but note that 32 − 8(12) = 1 9 N/A 10 $3 + \sqrt{10}$ 11 $10 + 3 \sqrt{11}$ 12 N/A, but 72 − 12(22) = 1 13 $18 + 5 \sqrt{13}$ 14 $15 + 4 \sqrt{14}$ 15 $4 + \sqrt{15}$ 16 N/A 17 $4 + \sqrt{17}$ 18 N/A, but 172 − 18(42) = 1 19 $170 + 39 \sqrt{19}$ 20 N/A, but 92 − 20(22) = 1 21 $55 + 12 \sqrt{21}$ 22 $197 + 42 \sqrt{22}$ 23 $24 + 5 \sqrt{23}$ 24 N/A, but 52 − 24(12) = 1 25 N/A 26 $5 + \sqrt{26}$ 27 N/A, but 262 − 27(52) = 1 28 N/A, but 1272 − 28(242) = 1 29 $70 + 13 \sqrt{29}$

Examples

• The units of $\scriptstyle \mathbb{N} \,$ are: $\scriptstyle \{1^0\} \,=\, \{1\} \,$ where $\scriptstyle 1 \,\equiv\, e^{{i 2 \pi}} \,$ (cf. natural numbers)
• The units of $\scriptstyle \mathbb{Z} \,$ are: $\scriptstyle \{(-1)^0, (-1)^1\} \,=\, \{1, -1\} \,$ where $\scriptstyle -1 \,\equiv\, e^{{i \pi}} \,$ (cf. rational integers)
• The units of $\scriptstyle \mathbb{Z}[\omega] \,$ are: $\scriptstyle \{\omega^0, \omega^1, \omega^2\} \,=\, \{1, \omega, \omega^2\} \,$ where $\scriptstyle \omega \,\equiv\, e^{\frac{i 2 \pi}{3}} \,$ (cf. Eisenstein integers)
• The units of $\scriptstyle \mathbb{Z}[i] \,$ are: $\scriptstyle \{i^0, i^1, i^2, i^3\} \,=\, \{1, i, -1, -i\} \,$ where $\scriptstyle i \,\equiv\, e^{\frac{i \pi}{2}} \,$ (cf. Gaussian integers)
• The units of $\scriptstyle \mathbb{Q} \,$ are: $\scriptstyle \mathbb{Q} \,$ \ $\scriptstyle \{0\} \,$
• The units of $\scriptstyle \mathbb{R} \,$ are: $\scriptstyle \mathbb{R} \,$ \ $\scriptstyle \{0\} \,$
• The units of $\scriptstyle \mathbb{C} \,$ are: $\scriptstyle \mathbb{C} \,$ \ $\scriptstyle \{0\} \,$