Golden field

== Golden integers ==

== Golden integers ==

[[File:Golden integer lattice.png |thumb |upright=1.5 |One convenient way to plot {{tmath|\Z[\varphi]}} is as a [[Lattice (group)|lattice]], using the number as the horizontal coordinate and its conjugate as the vertical coordinate. Then numbers with the same norm lie on [[hyperbola]]s (orange and green lines).]]

[[File:Golden integer lattice.png |thumb |upright=1.5 |One convenient way to plot {{tmath|\Z[\varphi]}} is as a [[Lattice (group)|lattice]], using the number as the horizontal coordinate and its conjugate as the vertical coordinate. Then numbers with the same norm lie on [[hyperbola]]s (orange and green lines).]]

The [[ring of integers]] of the golden field, {{tmath|\Z[\varphi]}}, sometimes called the '''golden integers''',<ref>For instance by {{harvnb|Rokhsar|Mermin|Wright|1987}}.</ref> is the set of numbers of the form <math>a + b\varphi</math> where <math>a</math> and <math>b</math> are both [[integer|ordinary integers]].{{sfnm|1a1=Hirzebruch|1y=1976 |2a1=Sporn|2y=2021}} This is the set of golden rational numbers whose norm is an integer. The set of all norms of golden integers includes every number {{tmath|1=\textstyle a^2 + ab - b^2}} for ordinary integers {{tmath|a}} and {{tmath|b}}. These are precisely the integers whose prime factors which are congruent to {{tmath|\pm 2}} [[Modular arithmetic|modulo]] {{tmath|5}} occur with even exponents. The first several non-negative norms are:<ref>{{harvnb|Sloane|at="Positive numbers of the form {{tmath|\textstyle x^2 + xy - y^2}}", OEIS {{OEIS link | A031363}} }}.</ref>

The [[ring of integers]] of the golden field, {{tmath|\Z[\varphi]}}, sometimes called the '''golden integers''',<ref>For instance by {{harvnb|Rokhsar|Mermin|Wright|1987}}.</ref> is the set of numbers of the form <math>a + b\varphi</math> where <math>a</math> and <math>b</math> are both [[integer|ordinary integers]].{{sfnm|1a1=Hirzebruch|1y=1976 |2a1=Sporn|2y=2021}} This is the set of numbers in {{Q(√5)}} whose norm is an integer. The set of all norms of golden integers includes every number {{tmath|1=\textstyle a^2 + ab - b^2}} for ordinary integers {{tmath|a}} and {{tmath|b}}. These are precisely the integers whose prime factors which are congruent to {{tmath|\pm 2}} [[Modular arithmetic|modulo]] {{tmath|5}} occur with even exponents. The first several non-negative norms are:<ref>{{harvnb|Sloane|at="Positive numbers of the form {{tmath|\textstyle x^2 + xy - y^2}}", OEIS {{OEIS link | A031363}} }}.</ref>

: {{tmath|0}}, {{tmath|1}}, {{tmath|4}}, {{tmath|5}}, {{tmath|9}}, {{tmath|11}}, {{tmath|16}}, {{tmath|19}}, {{tmath|20}}, {{tmath|25}}, {{tmath|29}},{{nobr| .{{hairsp}}.{{hairsp}}.{{thinsp}}.}}

: {{tmath|0}}, {{tmath|1}}, {{tmath|4}}, {{tmath|5}}, {{tmath|9}}, {{tmath|11}}, {{tmath|16}}, {{tmath|19}}, {{tmath|20}}, {{tmath|25}}, {{tmath|29}},{{nobr| .{{hairsp}}.{{hairsp}}.{{thinsp}}.}}

The golden integer {{tmath|0 + 0\varphi}} is called ''zero'', and is the only element of {{tmath|\Z[\varphi]}} with norm {{tmath|0}}.{{sfn|Dodd|1983|p=3}}

A ''[[unit (ring theory)|unit]]'' is an [[algebraic integer]] whose multiplicative inverse is also an algebraic integer, which happens when its norm is {{tmath|\pm 1}}. The units of {{Q(√5)}} are given by integer powers of the golden ratio and their negatives, {{tmath|\pm\varphi^n}}, for any integer {{tmath|n}}.{{sfn|Lind|1968}} Some [[Golden ratio#Golden ratio conjugate and powers|powers of {{tmath|\varphi}}]] are {{nobr|.{{hairsp}}.{{hairsp}}.}} {{tmath|1= \varphi^{-2} = 2 - \varphi}}, {{tmath|1= \varphi^{-1} = -1 + \varphi}}, {{tmath|1= \varphi^0 = 1}}, {{tmath|1= \varphi^1 = \varphi}}, {{tmath|1= \varphi^2 = 1 + \varphi}}, {{tmath|1= \varphi^3 = 1 + 2\varphi}}, {{nobr|.{{hairsp}}.{{hairsp}}.}} and in general {{tmath|1=\textstyle \varphi^{n} = F_{n-1} + F_n\varphi }}, where {{tmath|F_n}} is the {{tmath|n}}th [[Fibonacci number]].{{sfn|Dodd|1983|p=22}}{{sfn|Dimitrov|Cosklev|Bonevsky|1995}}

A ''[[unit (ring theory)|unit]]'' is an [[algebraic integer]] whose multiplicative inverse is also an algebraic integer, which happens when its norm is {{tmath|\pm 1}}. The units of {{Q(√5)}} are given by integer powers of the golden ratio and their negatives, {{tmath|\pm\varphi^n}}, for any integer {{tmath|n}}.{{sfn|Lind|1968}} Some [[Golden ratio#Golden ratio conjugate and powers|powers of {{tmath|\varphi}}]] are {{nobr|.{{hairsp}}.{{hairsp}}.}} {{tmath|1= \varphi^{-2} = 2 - \varphi}}, {{tmath|1= \varphi^{-1} = -1 + \varphi}}, {{tmath|1= \varphi^0 = 1}}, {{tmath|1= \varphi^1 = \varphi}}, {{tmath|1= \varphi^2 = 1 + \varphi}}, {{tmath|1= \varphi^3 = 1 + 2\varphi}}, {{nobr|.{{hairsp}}.{{hairsp}}.}} and in general {{tmath|1=\textstyle \varphi^{n} = F_{n-1} + F_n\varphi }}, where {{tmath|F_n}} is the {{tmath|n}}th [[Fibonacci number]].{{sfn|Dodd|1983|p=22}}{{sfn|Dimitrov|Cosklev|Bonevsky|1995}}