Golden field

In mathematics, ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠, sometimes called the golden field,^[1] is the real quadratic field obtained by extending the rational numbers with the square root of 5. The elements of this field are all of the numbers ⁠${\displaystyle a+b{\sqrt {5}}}$⁠, where ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ are both rational numbers. As a field, ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ supports the same basic arithmetical operations as the rational numbers. The name comes from the golden ratio ⁠${\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}}$⁠, which is the fundamental unit of ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠, and which satisfies the equation ⁠${\displaystyle \textstyle \varphi ^{2}=\varphi +1}$⁠.

Calculations in the golden field can be used to study the Fibonacci sequence and other topics related to the golden ratio, notably the geometry of the regular pentagon and higher-dimensional shapes with fivefold symmetry.

Basic arithmetic

Elements of the golden field can be written in the form ⁠${\displaystyle a+b{\sqrt {5}}}$⁠ where ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ are uniquely determined^[2] rational numbers, or in the form ⁠${\displaystyle {\bigl (}a+b{\sqrt {5}}~\!{\bigr )}{\big /}c}$⁠ where ⁠${\displaystyle a}$⁠, ⁠${\displaystyle b}$⁠, and ⁠${\displaystyle c}$⁠ are integers, which can be uniquely reduced to lowest terms. It is sometimes more convenient instead to use the form ⁠${\displaystyle a+b\varphi }$⁠ where ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ are rational and ⁠${\displaystyle \varphi }$⁠ is the golden ratio, or the form ⁠${\displaystyle (a+b\varphi )/c}$⁠ where ⁠${\displaystyle a}$⁠, ⁠${\displaystyle b}$⁠, and ⁠${\displaystyle c}$⁠ are integers.^[3]

Converting between these alternative forms is straight-forward: ⁠${\displaystyle a+b{\sqrt {5}}=(a-b)+(2b)\varphi }$⁠, or in the other direction ⁠${\displaystyle a+b\varphi ={\bigl (}a+{\tfrac {1}{2}}b{\bigr )}+{\bigl (}{\tfrac {1}{2}}b{\bigr )}{\sqrt {5}}}$⁠.^[4]

To add or subtract two numbers, simply add or subtract the components separately:^[5] $${\displaystyle {\begin{aligned}{\bigl (}a_{1}+b_{1}{\sqrt {5}}~\!{\bigr )}+{\bigl (}a_{2}+b_{2}{\sqrt {5}}~\!{\bigr )}&=(a_{1}+a_{2})+(b_{1}+b_{2}){\sqrt {5}},\\[3mu](a_{1}+b_{1}\varphi )+(a_{2}+b_{2}\varphi )&=(a_{1}+a_{2})+(b_{1}+b_{2})\varphi .\end{aligned}}}$$

To multiply two numbers, distribute:^[5] $${\displaystyle {\begin{aligned}{\bigl (}a_{1}+b_{1}{\sqrt {5}}~\!{\bigr )}{\bigl (}a_{2}+b_{2}{\sqrt {5}}~\!{\bigr )}&=(a_{1}a_{2}+5b_{1}b_{2})+(a_{1}b_{2}+b_{1}a_{2}){\sqrt {5}},\\[3mu](a_{1}+b_{1}\varphi )(a_{2}+b_{2}\varphi )&=(a_{1}a_{2}+b_{1}b_{2})+(a_{1}b_{2}+b_{1}a_{2}+b_{1}b_{2})\varphi .\end{aligned}}}$$

To find the reciprocal of a number ${\displaystyle \alpha }$, rationalize the denominator: ${\displaystyle 1/\alpha ={}}$${\displaystyle {\bar {\alpha }}/\alpha {\bar {\alpha }}={}}$${\displaystyle {\bar {\alpha }}/\mathrm {N} (\alpha )}$, where ⁠${\displaystyle {\bar {\alpha }}}$⁠ is the conjugate and ⁠${\displaystyle \mathrm {N} (\alpha )}$⁠ is the field norm, as defined below. Explicitly:

$${\displaystyle {\begin{aligned}{\frac {1}{a+b{\sqrt {5}}}}&={\frac {a}{a^{2}-5b^{2}}}-{\frac {b}{a^{2}-5b^{2}}}{\sqrt {5}},\\[3mu]{\frac {1}{a+b\varphi }}&={\frac {a+b}{a^{2}+ab-b^{2}}}-{\frac {b}{a^{2}+ab-b^{2}}}\varphi .\end{aligned}}}$$

To divide two numbers, multiply the first by second's reciprocal, ${\displaystyle \alpha _{1}/\alpha _{2}={}}$${\displaystyle \alpha _{1}{\bar {\alpha }}_{2}/\mathrm {N} (\alpha _{2})}$. Explicitly: $${\displaystyle {\begin{aligned}{\frac {a_{1}+b_{1}{\sqrt {5}}}{a_{2}+b_{2}{\sqrt {5}}}}&={\frac {a_{1}a_{2}-5b_{1}b_{2}}{a_{2}^{2}-5b_{2}^{2}}}+{\frac {-a_{1}b_{2}+b_{1}a_{2}}{a_{2}^{2}-5b_{2}^{2}}}{\sqrt {5}},\\[6mu]{\frac {a_{1}+b_{1}\varphi }{a_{2}+b_{2}\varphi }}&={\frac {a_{1}a_{2}+a_{1}b_{2}-b_{1}b_{2}}{a_{2}^{2}+a_{2}b_{2}-b_{2}^{2}}}+{\frac {-a_{1}b_{2}+b_{1}a_{2}}{a_{2}^{2}+a_{2}b_{2}-b_{2}^{2}}}\varphi .\end{aligned}}}$$

Conjugation and norm

The numbers ⁠${\displaystyle {\sqrt {5}}}$⁠ and ⁠${\displaystyle -{\sqrt {5}}}$⁠ each solve the equation ⁠${\displaystyle \textstyle x^{2}=5}$⁠. Each number ⁠${\displaystyle a+b{\sqrt {5}}}$⁠ in ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ has an algebraic conjugate found by swapping these two square roots of 5, i.e., by changing the sign of ⁠${\displaystyle b}$⁠. The conjugate of ⁠${\displaystyle \varphi }$⁠ is ${\displaystyle {\bar {\varphi }}={\tfrac {1}{2}}{\bigl (}1-{\sqrt {5}}~\!{\bigr )}={}}$${\displaystyle \textstyle -\varphi ^{-1}={}}$${\displaystyle 1-\varphi }$. The conjugate of a number ⁠${\displaystyle x}$⁠ is commonly denoted ⁠${\displaystyle {\bar {x}}}$⁠, as with the complex conjugate. In general, the conjugate is: $${\displaystyle {\begin{aligned}{\overline {a+b{\sqrt {5}}}}&=a-b{\sqrt {5}},\\[3mu]{\overline {a+b\varphi }}&=a+b-b\varphi .\end{aligned}}}$$

The sum of a number and its conjugates is called the field trace or just the trace (so-called because multiplication by an element in the field can be seen as a kind of linear transformation, the trace of whose matrix is the field trace). The field trace of ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ is: $${\displaystyle {\begin{aligned}\mathrm {Tr} {\bigl (}a+b{\sqrt {5}}~\!{\bigr )}&={\bigl (}a+b{\sqrt {5}}~\!{\bigr )}+{\bigl (}a-b{\sqrt {5}}~\!{\bigr )}=2a,\\[3mu]\mathrm {Tr} (a+b\varphi )&=(a+b\varphi )+(a+b-b\varphi )=2a+b.\end{aligned}}}$$ This is always an (ordinary) rational number.

Multiplying a number in ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ by its conjugate gives a measure of that number's "size" or "magnitude", called the field norm or just the norm.^[6] The field norm of ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ is: $${\displaystyle {\begin{aligned}\mathrm {N} {\bigl (}a+b{\sqrt {5}}~\!{\bigr )}&=a^{2}-5b^{2},\\[3mu]\mathrm {N} (a+b\varphi )&=a^{2}+ab-b^{2}.\end{aligned}}}$$ This is also always a rational number.

The norm has some properties expected for magnitudes. For instance, a number and its conjugate have the same norm, ⁠${\displaystyle \mathrm {N} (\alpha )=\mathrm {N} ({\bar {\alpha }})}$⁠; the norm of a product is the product of norms, ⁠${\displaystyle \operatorname {N} (\alpha \beta )=\mathrm {N} (\alpha )\mathrm {N} (\beta )}$⁠; and the norm of a number's reciprocal is the reciprocal of the norm, ⁠${\displaystyle \mathrm {N} (1/\alpha )=1/\mathrm {N} (\alpha )}$⁠.^[7]

A number ⁠${\displaystyle \alpha }$⁠ in ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ and its conjugate ⁠${\displaystyle {\bar {\alpha }}}$⁠ are the solutions of the quadratic equation^[7] $${\displaystyle (x-\alpha )(x-{\bar {\alpha }})=x^{2}-\mathrm {Tr} (\alpha )x+\mathrm {N} (\alpha )=0.}$$

In Galois theory, the golden field can be considered more abstractly as the set of all numbers ⁠${\displaystyle a+bu}$⁠, where ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ are both rational, and all that is known of ⁠${\displaystyle u}$⁠ is that it satisfies the equation ⁠${\displaystyle \textstyle u^{2}=5}$⁠. There are two ways to embed this set in the real numbers: by mapping ⁠${\displaystyle u}$⁠ to the positive square root ⁠${\displaystyle {\sqrt {5}}}$⁠ or alternatively by mapping ⁠${\displaystyle u}$⁠ to the negative square root ⁠${\displaystyle -{\sqrt {5}}}$⁠. Conjugation exchanges these two embeddings. The Galois group of the golden field is thus the group with two elements, namely the identity and an element which is its own inverse.^[6]

Golden integers

One convenient way to plot ⁠${\displaystyle \mathbb {Z} [\varphi ]}$⁠ is as a lattice, using the number as the horizontal coordinate and its conjugate as the vertical coordinate. Then numbers with the same norm lie on hyperbolas (orange and green lines).

The ring of integers of the golden field, ⁠${\displaystyle \mathbb {Z} [\varphi ]}$⁠, sometimes called the golden integers,^[8] is the set of numbers of the form ${\displaystyle a+b\varphi }$ where ${\displaystyle a}$ and ${\displaystyle b}$ are both ordinary integers.^[9] This is the set of golden rational numbers whose norm is an integer. A unit is an algebraic integer whose multiplicative inverse is also an algebraic integer, which happens when its norm is ⁠${\displaystyle \pm 1}$⁠. The units of ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ are given by integer powers of the golden ratio and their negatives, ⁠${\displaystyle \pm \varphi ^{n}}$⁠, for any integer ⁠${\displaystyle n}$⁠.^[2] Some powers of ⁠${\displaystyle \varphi }$⁠ are ... ⁠${\displaystyle \varphi ^{-2}=2-\varphi }$⁠, ⁠${\displaystyle \varphi ^{-1}=-1+\varphi }$⁠, ⁠${\displaystyle \varphi ^{0}=1}$⁠, ⁠${\displaystyle \varphi ^{1}=\varphi }$⁠, ⁠${\displaystyle \varphi ^{2}=1+\varphi }$⁠, ⁠${\displaystyle \varphi ^{3}=1+2\varphi }$⁠, ... and in general ⁠${\displaystyle \textstyle \varphi ^{n}=F_{n-1}+F_{n}\varphi }$⁠, where ⁠${\displaystyle F_{n}}$⁠ is the ⁠${\displaystyle n}$⁠th Fibonacci number.^[5]

Golden integer units (hollow circles) and primes (filled circles), along with zero (+) and composite numbers (×)^[10]

The prime elements of the ring, analogous to prime numbers among the integers, are of three types: ⁠${\displaystyle {\sqrt {5}}=-1+2\varphi }$⁠, integer primes of the form ⁠${\displaystyle p=5n\pm 2}$⁠ where ⁠${\displaystyle n}$⁠ is an integer, and the factors of integer primes of the form ⁠${\displaystyle p=5n\pm 1}$⁠ (a pair of conjugates).^[11] For example, ⁠${\displaystyle 2}$⁠, ⁠${\displaystyle 3}$⁠, and ⁠${\displaystyle 7}$⁠ are primes, but ⁠${\displaystyle 11=(3+\varphi )(4-\varphi )}$⁠ is composite. Any of these is an associate of additional primes found by multiplying it by a unit; for example ⁠${\displaystyle 2\varphi }$⁠ is also prime because ⁠${\displaystyle \varphi }$⁠ is a unit.

The ring ⁠${\displaystyle \mathbb {Z} [\varphi ]}$⁠ is a Euclidean domain with the absolute value of the norm as its Euclidean function, meaning a version of the Euclidean algorithm can be used to find the greatest common divisor of two numbers.^[12] This makes ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ one of the 21 quadratic fields that are norm-Euclidean.^[13]

Like all Euclidean domains, the ring ⁠${\displaystyle \mathbb {Z} [\varphi ]}$⁠ shares many properties with the ring of integers. In particular, it is a principal ideal domain, and it satisfies a form of the fundamental theorem of arithmetic: every element of ⁠${\displaystyle \mathbb {Z} [\varphi ]}$⁠ can be written as a product of prime elements multiplied by a unit, and this factorization is unique up to the order of the factors and the replacement of any prime factor by an associate prime (which changes the unit factor accordingly).

Other properties

The golden field is the real quadratic field with the smallest discriminant.^[14] It has class number 1 and is a unique factorization domain.^[15]

Any positive element of the golden field can be written as a generalized type of continued fraction, in which the partial quotients are sums of non-negative powers of ${\displaystyle \varphi }$.^[16]

Relation to fivefold symmetry

The golden ratio ⁠${\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}}$⁠ is the ratio between the lengths of a diagonal and a side of a regular pentagon, so the golden field and golden integers feature prominently in the metrical geometry of the regular pentagon and its symmetry system, as well as higher-dimensional objects and symmetries involving five-fold symmetry.

Euclidean plane

The golden ratio is related to the fifth roots of unity.

Let ⁠${\displaystyle \zeta ={\exp }{\bigl (}2\pi i/5{\bigr )}}$⁠ be the 5th root of unity, a complex number of unit absolute value spaced ⁠${\displaystyle {\tfrac {1}{5}}}$⁠ of a full turn from ⁠${\displaystyle 1}$⁠ around the unit circle, satisfying ⁠${\displaystyle \textstyle \zeta ^{5}=1}$⁠. Then the fifth cyclotomic field ⁠${\displaystyle \mathbb {Q} (\zeta )}$⁠ is the field extension of the rational numbers formed by adjoining ⁠${\displaystyle \zeta }$⁠ (or equivalently, adjoining any of ⁠${\displaystyle \textstyle \zeta ^{2}}$⁠, ⁠${\displaystyle \textstyle \zeta ^{3}}$⁠ or ⁠${\displaystyle \textstyle \zeta ^{4}}$⁠). Elements of ⁠${\displaystyle \mathbb {Q} (\zeta )}$⁠ are numbers of the form ⁠${\displaystyle \textstyle a_{0}+a_{1}\zeta +a_{2}\zeta ^{2}+a_{3}\zeta ^{3}+a_{4}\zeta ^{4}}$⁠, with rational coefficients. ⁠${\displaystyle \mathbb {Q} (\zeta )}$⁠ is of degree four over the rational numbers: any four of the five roots are linearly independent over ⁠${\displaystyle \mathbb {Q} }$⁠, but they sum to the additive inverse of the fifth. However, ⁠${\displaystyle \mathbb {Q} (\zeta )}$⁠ is only of degree two over ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠, $${\displaystyle {\begin{aligned}x^{5}-1&=(x-1){\bigl (}x^{4}+x^{3}+x^{2}+x+1{\bigr )}\\[2mu]&=(x-1){\bigl (}x^{2}+{\bar {\varphi }}x+1{\bigr )}{\bigl (}x^{2}+\varphi x+1{\bigr )}\\[2mu]&=(x-1){\bigl (}x-\zeta {\bigr )}{\bigl (}x-\zeta ^{4}{\bigr )}{\bigl (}x-\zeta ^{2}{\bigr )}{\bigl (}x-\zeta ^{3}{\bigr )},\end{aligned}}}$$ where the conjugate ⁠${\displaystyle {\bar {\varphi }}=1-\varphi }$⁠. The elements of ⁠${\displaystyle \mathbb {Q} (\zeta )}$⁠ can alternately be represented as ⁠${\displaystyle \alpha +\beta \zeta }$⁠, where ⁠${\displaystyle \alpha }$⁠ and ⁠${\displaystyle \beta }$⁠ are elements of ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠: $${\displaystyle {\begin{aligned}&a_{0}+a_{1}\zeta +a_{2}\zeta ^{2}+a_{3}\zeta ^{3}+a_{4}\zeta ^{4}\\[3mu]&\qquad ={\bigl (}a_{0}-a_{2}+{\bar {\varphi }}a_{3}-{\bar {\varphi }}a_{4}{\bigr )}+{\bigl (}a_{1}-{\bar {\varphi }}a_{2}+{\bar {\varphi }}a_{3}-a_{4}{\bigr )}\zeta .\end{aligned}}}$$

Conversely, ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ is a subfield of ⁠${\displaystyle \mathbb {Q} (\zeta )}$⁠. For any primitive root of unity ⁠${\displaystyle \zeta _{n}}$⁠, the maximal real subfield of the cyclotomic field ⁠${\displaystyle \mathbb {Q} (\zeta _{n})}$⁠ is the field ⁠${\displaystyle \mathbb {Q} (\zeta _{n}+\zeta _{n}^{-1})}$⁠; see Minimal polynomial of ⁠${\displaystyle 2\cos(2\pi /n)}$⁠. In our case ⁠${\displaystyle n=5}$⁠, ⁠${\displaystyle \zeta +\zeta ^{-1}=\varphi ^{-1}}$⁠, so the maximal real subfield of ⁠${\displaystyle \mathbb {Q} (\zeta )}$⁠ is ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠.^[17] Golden integers are therefore involved in the trigonometric study of fivefold symmetries.

Three-dimensional space

Four-dimensional space

The vertices of the four-dimensional polytope known as the 600-cell can be chosen so that their coordinates lie within the golden integers.^[18]

Higher dimensions

The icosians are a special set of quaternions that are used in a construction of the E₈ lattice. Each component of an icosian always belongs to the golden field.^[19] The icosians of unit norm are the vertices of a 600-cell.^[18]

Quasiperiodicity

Golden integers are used in studying quasicrystals.^[20]

Other applications

The quintic case of Fermat's Last Theorem, that there are no nontrivial integer solutions to the equation ⁠${\displaystyle \textstyle a^{5}+b^{5}=c^{5}}$⁠, was proved using ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ by Gustav Lejeune Dirichlet and Adrien-Marie Legendre in 1825–1830.^[21]

In enumerative geometry, it is proven that every non-singular cubic surface contains exactly 27 lines. The Clebsch surface is unusual in that all 27 lines can be defined over the real numbers.^[22] They can, in fact, be defined over the golden field.^[23]

In quantum information theory, an abelian extension of the golden field is used in a construction of a SIC-POVM in four-dimensional complex vector space.^[24]