Golden field

In mathematics, ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠,^[1] sometimes called the golden field,^[2] 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 ⁠${\displaystyle \varphi }$⁠, 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 numbers 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 are those numbers which can be written in the form ⁠${\displaystyle a+b{\sqrt {5}}}$⁠ where ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ are uniquely determined^[3] 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, and where ⁠${\displaystyle {\sqrt {5}}=2.236\ldots }$⁠ is the square root of 5.^[4] It is sometimes more convenient instead to use the form ⁠${\displaystyle a+b\varphi }$⁠ where ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ are rational or the form ⁠${\displaystyle (a+b\varphi )/c}$⁠ where ⁠${\displaystyle a}$⁠, ⁠${\displaystyle b}$⁠, and ⁠${\displaystyle c}$⁠ are integers, and where ${\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}={}\!}$${\displaystyle 1.618\ldots }$ is the golden ratio.^[5][6]

Converting between these alternative forms is straight-forward: ⁠${\displaystyle a+b{\sqrt {5}}=(a-b)+(2b)\varphi }$⁠, or in the other direction ⁠${\displaystyle a+b\varphi ={\bigl (}a+{\tfrac {1}{2}}b{\bigr )}+{\bigl (}{\tfrac {1}{2}}b{\bigr )}{\sqrt {5}}}$⁠.^[7]

To add or subtract two numbers, simply add or subtract the components separately:^[8] $${\displaystyle {\begin{aligned}{\bigl (}a_{1}+b_{1}{\sqrt {5}}~\!{\bigr )}+{\bigl (}a_{2}+b_{2}{\sqrt {5}}~\!{\bigr )}&=(a_{1}+a_{2})+(b_{1}+b_{2}){\sqrt {5}},\\[3mu](a_{1}+b_{1}\varphi )+(a_{2}+b_{2}\varphi )&=(a_{1}+a_{2})+(b_{1}+b_{2})\varphi .\end{aligned}}}$$

To multiply two numbers, distribute:^[8] $${\displaystyle {\begin{aligned}{\bigl (}a_{1}+b_{1}{\sqrt {5}}~\!{\bigr )}{\bigl (}a_{2}+b_{2}{\sqrt {5}}~\!{\bigr )}&=(a_{1}a_{2}+5b_{1}b_{2})+(a_{1}b_{2}+b_{1}a_{2}){\sqrt {5}},\\[3mu](a_{1}+b_{1}\varphi )(a_{2}+b_{2}\varphi )&=(a_{1}a_{2}+b_{1}b_{2})+(a_{1}b_{2}+b_{1}a_{2}+b_{1}b_{2})\varphi .\end{aligned}}}$$

To find the reciprocal of a number ${\displaystyle \alpha }$, rationalize the denominator: ${\displaystyle 1/\alpha ={}}$${\displaystyle {\overline {\alpha }}/\alpha {\overline {\alpha }}={}}$${\displaystyle {\overline {\alpha }}/\mathrm {N} (\alpha )}$, where ⁠${\displaystyle {\overline {\alpha }}}$⁠ is the algebraic conjugate and ⁠${\displaystyle \mathrm {N} (\alpha )}$⁠ is the field norm, as defined below.^[9] 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}={}}$${\displaystyle \alpha _{1}{\overline {\alpha }}_{2}/\mathrm {N} (\alpha _{2})}$.^[9] Explicitly: $${\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 ⁠${\displaystyle \alpha =a+b{\sqrt {5}}}$⁠ in ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ has an algebraic conjugate ⁠${\displaystyle {\overline {\alpha }}}$⁠ 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 {\overline {\varphi }}={\tfrac {1}{2}}{\bigl (}1-{\sqrt {5}}~\!{\bigr )}={}}$${\displaystyle \textstyle -\varphi ^{-1}={}}$${\displaystyle 1-\varphi }$. A rational number is its own conjugate. In general, the conjugate is:^[10] $${\displaystyle {\begin{aligned}{\overline {a+b{\sqrt {5}}}}&=a-b{\sqrt {5}},\\[3mu]{\overline {a+b\varphi }}&=a+b{\overline {\varphi }}=(a+b)-b\varphi .\end{aligned}}}$$ Conjugation in ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ is an involution, ⁠${\displaystyle {\overline {({\overline {\alpha }})}}=\alpha }$⁠, and it preserves the structure of arithmetic: ⁠${\displaystyle {\overline {\alpha _{1}+\alpha _{2}}}={\overline {\alpha }}_{1}+{\overline {\alpha }}_{2}}$⁠; ⁠${\displaystyle {\overline {\alpha _{1}\alpha _{2}}}={\overline {\alpha }}_{1}{\overline {\alpha }}_{2}}$⁠; and ⁠${\displaystyle {\overline {\alpha _{1}/\alpha _{2}}}={\overline {\alpha }}_{1}/\,{\overline {\alpha }}_{2}}$⁠.^[11] Conjugation is the only ring homomorphism (function preserving the structure of addition and multiplication) from ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ to itself, other than the identity function.^[12]

The field trace is the sum of a number and its conjugates (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; see § Matrix representation).^[13] The trace of ⁠${\displaystyle \alpha }$⁠ in ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ is ⁠${\displaystyle \mathrm {Tr} (\alpha )=\alpha +{\overline {\alpha }}}$⁠: $${\displaystyle {\begin{aligned}\mathrm {Tr} {\bigl (}a+b{\sqrt {5}}~\!{\bigr )}&={\bigl (}a+b{\sqrt {5}}~\!{\bigr )}+{\bigl (}a-b{\sqrt {5}}~\!{\bigr )}=2a,\\[3mu]\mathrm {Tr} (a+b\varphi )&=(a+b\varphi )+(a+b-b\varphi )=2a+b.\end{aligned}}}$$ This is always an (ordinary) rational number.^[11]

The field norm is a measure of a number's magnitude, the product of the number and its conjugates.^[14] The norm of ⁠${\displaystyle \alpha }$⁠ in ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ is ⁠${\displaystyle \mathrm {N} (\alpha )=\alpha {\overline {\alpha }}}$⁠:^[11] $${\displaystyle {\begin{aligned}\mathrm {N} {\bigl (}a+b{\sqrt {5}}~\!{\bigr )}&=a^{2}-5b^{2},\\[3mu]\mathrm {N} (a+b\varphi )&=a^{2}+ab-b^{2}.\end{aligned}}}$$ This is also always a rational number.^[11]

The norm has some properties expected for magnitudes. For instance, a number and its conjugate have the same norm, ⁠${\displaystyle \mathrm {N} (\alpha )=\mathrm {N} ({\overline {\alpha }})}$⁠; the norm of a product is the product of norms, ⁠${\displaystyle \operatorname {N} (\alpha _{1}\alpha _{2})=\mathrm {N} (\alpha _{1})~\!\mathrm {N} (\alpha _{2})}$⁠; and the norm of a quotient is the quotient of the norms, ⁠${\displaystyle \mathrm {N} (\alpha _{1}/\alpha _{2})=\mathrm {N} (\alpha _{1}){\big /}~\!\mathrm {N} (\alpha _{2})}$⁠.^[11]

A number ⁠${\displaystyle \alpha }$⁠ in ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ and its conjugate ⁠${\displaystyle {\overline {\alpha }}}$⁠ are the solutions of the quadratic equation^[11] $${\displaystyle (x-\alpha )(x-{\overline {\alpha }})=x^{2}-\mathrm {Tr} (\alpha )x+\mathrm {N} (\alpha )=0.}$$

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.^[14]

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,^[15] is the subset of algebraic integers in the field, meaning those elements whose minimal polynomial over ⁠${\displaystyle \mathbb {Q} }$⁠ has integer coefficients. These are the set of numbers in ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ whose norm is an integer. The numbers ⁠${\displaystyle 1}$⁠ and ⁠${\displaystyle \varphi }$⁠ form an integral basis for the ring, meaning every number in the ring can be written in the form ${\displaystyle a+b\varphi }$ where ${\displaystyle a}$ and ${\displaystyle b}$ are ordinary integers.^[16] Alternately, elements of ⁠${\displaystyle \mathbb {Z} [\varphi ]}$⁠ can be written in the form ⁠${\displaystyle {\tfrac {1}{2}}a+{\tfrac {1}{2}}b{\sqrt {5}}}$⁠, where ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ have the same parity.^[17] Like any ring, ⁠${\displaystyle \mathbb {Z} [\varphi ]}$⁠ is closed under addition and multiplication.

The set of all norms of golden integers includes every number ${\displaystyle \textstyle a^{2}+ab-b^{2}={}\!}$${\displaystyle \mathrm {N} (a+b\varphi )}$ for ordinary integers ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠. These are precisely the integers whose prime factors which are congruent to ⁠${\displaystyle \pm 2}$⁠ modulo ⁠${\displaystyle 5}$⁠ occur with even exponents. The first several non-negative integer norms are:^[18]

⁠${\displaystyle 0}$⁠, ⁠${\displaystyle 1}$⁠, ⁠${\displaystyle 4}$⁠, ⁠${\displaystyle 5}$⁠, ⁠${\displaystyle 9}$⁠, ⁠${\displaystyle 11}$⁠, ⁠${\displaystyle 16}$⁠, ⁠${\displaystyle 19}$⁠, ⁠${\displaystyle 20}$⁠, ⁠${\displaystyle 25}$⁠, ⁠${\displaystyle 29}$⁠, . . . .

The golden integer ⁠${\displaystyle 0+0\varphi }$⁠ is called zero, and is the only element of ⁠${\displaystyle \mathbb {Z} [\varphi ]}$⁠ with norm ⁠${\displaystyle 0}$⁠.^[19]

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 ⁠${\displaystyle n}$⁠.^[3] 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.^[20][8]

Golden integer units (hollow circles) and primes (filled circles), along with zero (+) and composite numbers (×)^[21]

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).^[22] 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.^[23] This makes ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ one of the 21 quadratic fields that are norm-Euclidean.^[24]

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).

Matrix representation

⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ is a two-dimensional vector space over ⁠${\displaystyle \mathbb {Q} }$⁠, and multiplication by any element of ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ is a linear transformation of that vector space. Thus the arithmetic of numbers in ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ can alternately be represented using the arithmetic of a particular set of ⁠${\displaystyle 2\times 2}$⁠ square matrices with rational entries, spanned by an identity matrix ⁠${\displaystyle \mathbf {I} }$⁠ representing the number ⁠${\displaystyle 1}$⁠ and any matrix ⁠${\displaystyle \mathbf {\Phi } }$⁠ with characteristic polynomial ⁠${\displaystyle \textstyle x^{2}-x-1}$⁠ representing the number ⁠${\displaystyle \varphi }$⁠.^[25] In general the matrix ⁠${\displaystyle \mathbf {A} =a\mathbf {I} +b\mathbf {\Phi } }$⁠ represents the number ⁠${\displaystyle \alpha =a+b\varphi }$⁠.^[26] A convenient choice of ⁠${\displaystyle \mathbf {\Phi } }$⁠ is the symmetric matrix:^[27] $${\displaystyle {\begin{aligned}&\mathbf {I} ={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\quad \mathbf {\Phi } ={\begin{bmatrix}0&1\\1&1\end{bmatrix}},\quad \mathbf {A} =a\mathbf {I} +b\mathbf {\Phi } ={\begin{bmatrix}a&b\\b&a+b\end{bmatrix}}.\quad \end{aligned}}}$$

The adjugate matrix ⁠${\displaystyle {\overline {\mathbf {\Phi } }}=\mathbf {I} -\mathbf {\Phi } }$⁠ represents the algebraic conjugate ⁠${\displaystyle {\overline {\varphi }}=1-\varphi }$⁠, a matrix ⁠${\displaystyle \mathbf {R} =-\mathbf {I} +2\mathbf {\Phi } }$⁠ (satisfying ⁠${\displaystyle \textstyle \mathbf {R} ^{2}=5\mathbf {I} }$⁠) represents ⁠${\displaystyle {\sqrt {5}}}$⁠,^[28] and the adjugate of an arbitrary element ⁠${\displaystyle \mathbf {A} }$⁠, which we will denote ⁠${\displaystyle {\overline {\mathbf {A} }}=a\mathbf {I} +b{\overline {\mathbf {\Phi } }}}$⁠, represents the number ⁠${\displaystyle {\overline {\alpha }}=a+b{\overline {\varphi }}}$⁠:

$${\displaystyle {\begin{aligned}&\mathbf {R} ={\begin{bmatrix}-1&2\\2&1\end{bmatrix}},\quad {\overline {\mathbf {\Phi } }}={\begin{bmatrix}~1&-1\\-1&\,0\end{bmatrix}},\quad {\overline {\mathbf {A} }}=a\mathbf {I} +b{\overline {\mathbf {\Phi } }}={\begin{bmatrix}a+b&-b~\\-b&~a\end{bmatrix}}.\end{aligned}}}$$

If ⁠${\displaystyle \alpha =a+b\varphi }$⁠ is an element of ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠, with conjugate ⁠${\displaystyle {\overline {\alpha }}=a+b{\overline {\varphi }}}$⁠, then the matrix ⁠${\displaystyle \mathbf {A} =a\mathbf {I} +b\mathbf {\Phi } }$⁠ has the numbers ⁠${\displaystyle \alpha }$⁠ and ⁠${\displaystyle {\overline {\alpha }}}$⁠ as its eigenvalues. Its trace is ${\displaystyle \mathrm {Tr} (\mathbf {A} )={}\!}$${\displaystyle 2a+b={}\!}$${\displaystyle \mathrm {Tr} (\alpha )}$. Its determinant is ${\displaystyle \mathrm {det} (\mathbf {A} )={}\!}$${\displaystyle \textstyle a^{2}+ab-b^{2}={}\!}$${\displaystyle \mathrm {N} (\alpha )}$. The characteristic polynomial of ⁠${\displaystyle \mathbf {A} }$⁠ is ⁠${\displaystyle (x-\alpha )(x-{\overline {\alpha }})}$⁠, which is the minimal polynomial of ⁠${\displaystyle \alpha }$⁠ and ⁠${\displaystyle {\overline {\alpha }}}$⁠ whenever ⁠${\displaystyle b}$⁠ is not zero. These properties are shared by the adjugate matrix ⁠${\displaystyle {\overline {\mathbf {A} }}}$⁠. Their product is ⁠${\displaystyle \mathbf {A} {\overline {\mathbf {A} }}=\mathrm {det} (\mathbf {A} )\mathbf {I} }$⁠.^[25][29]

Matrix arithmetic with matrices of the form ⁠${\displaystyle \mathbf {A} =a\mathbf {I} +b\mathbf {\Phi } }$⁠ is isomorphic to arithmetic in ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠.^[29] Every such matrix ⁠${\displaystyle \mathbf {A} }$⁠, except for the zero matrix, is invertible, and its inverse ⁠${\displaystyle \textstyle \mathbf {A} ^{-1}={\bigl (}1/\mathrm {det} (\mathbf {A} ){\bigr )}{\overline {\mathbf {A} }}}$⁠ represents the multiplicative inverse ⁠${\displaystyle \textstyle \alpha ^{-1}={\overline {\alpha }}/\mathrm {N} (\alpha )}$⁠ in ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠.^[30]

These matrices have especially been studied in the context of the Fibonacci numbers ⁠${\displaystyle F_{n}}$⁠ and Lucas numbers ⁠${\displaystyle L_{n}}$⁠, which appear as the entries of ⁠${\displaystyle \textstyle \mathbf {\Phi } ^{n}}$⁠ and ⁠${\displaystyle \textstyle \mathbf {\Phi } ^{n}\mathbf {R} }$⁠, respectively: $${\displaystyle {\begin{aligned}\mathbf {\Phi } ^{n}&=F_{n-1}\mathbf {I} +F_{n}\mathbf {\Phi } ={\begin{bmatrix}F_{n-1}&F_{n}~\\F_{n}~&F_{n+1}\end{bmatrix}},&\mathrm {Tr} (\mathbf {\Phi } ^{n})&=L_{n},\\[8mu]\mathbf {\Phi } ^{n}\mathbf {R} &=L_{n-1}\mathbf {I} +L_{n}\mathbf {\Phi } ={\begin{bmatrix}L_{n-1}&L_{n}~\\L_{n}~&L_{n+1}\end{bmatrix}},&\mathrm {Tr} (\mathbf {\Phi } ^{n}\mathbf {R} )&=5F_{n}.\end{aligned}}}$$ Powers of ⁠${\displaystyle \mathbf {\Phi } }$⁠ are sometimes called Fibonacci matrices.^[31]

Other properties

The golden field is the real quadratic field with the smallest discriminant, ⁠${\displaystyle \Delta _{\mathbb {Q} \left(~\!\!{\sqrt {5}}\right)}=5}$⁠.^[32] It has class number 1 and is a unique factorization domain.^[33]

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 ${\displaystyle \varphi }$.^[34]

Fibonacci numbers

The Lucas and Fibonacci numbers are components of φⁿ when written in terms of ⁠1/2⁠ and ⁠1/2⁠√5.^[35]

Binet's formula for Fibonacci numbers plotted in the lattice of golden integers

⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠ is the natural number system to use when studying the Fibonacci numbers ⁠${\displaystyle F_{n}}$⁠ and the Lucas numbers ⁠${\displaystyle L_{n}}$⁠. These number sequences are usually defined by recurrence relations similar to the one satisfied by the powers of ⁠${\displaystyle \varphi }$⁠ and ⁠${\displaystyle {\overline {\varphi }}}$⁠: $${\displaystyle {\begin{aligned}F_{0}&=0,&F_{1}&=1,&F_{n+1}&=F_{n}+F_{n-1},\\[3mu]L_{0}&=2,&L_{1}&=1,&L_{n+1}&=L_{n}+L_{n-1},\\[3mu]\varphi ^{0}&=1,&\varphi ^{1}&=\varphi ,&\varphi ^{n+1}&=\varphi ^{n}+\varphi ^{n-1},\\[3mu]{\overline {\varphi }}^{0}&=1,&{\overline {\varphi }}^{1}&={\overline {\varphi }},&{\overline {\varphi }}^{n+1}&={\overline {\varphi }}^{n}+{\overline {\varphi }}^{n-1}.\end{aligned}}}$$

The sequences ⁠${\displaystyle F_{n}}$⁠ and ⁠${\displaystyle L_{n}}$⁠ respectively begin:^[36]

⁠${\displaystyle 0}$⁠, ⁠${\displaystyle 1}$⁠, ⁠${\displaystyle 1}$⁠, ⁠${\displaystyle 2}$⁠, ⁠${\displaystyle 3}$⁠, ⁠${\displaystyle 5}$⁠, ⁠${\displaystyle 8}$⁠, ⁠${\displaystyle 13}$⁠, ⁠${\displaystyle 21}$⁠, ⁠${\displaystyle 34}$⁠, ⁠${\displaystyle 55}$⁠, . . . ;

⁠${\displaystyle 2}$⁠, ⁠${\displaystyle 1}$⁠, ⁠${\displaystyle 3}$⁠, ⁠${\displaystyle 4}$⁠, ⁠${\displaystyle 7}$⁠, ⁠${\displaystyle 11}$⁠, ⁠${\displaystyle 18}$⁠, ⁠${\displaystyle 29}$⁠, ⁠${\displaystyle 47}$⁠, ⁠${\displaystyle 76}$⁠, ⁠${\displaystyle 123}$⁠, . . . .

Both sequences can be consistently extended to negative integer indices by following the same recurrence in the negative direction. They satisfy the identities^[37] $${\displaystyle {\begin{aligned}F_{-n}&=(-1)^{n+1}F_{n},\\L_{-n}&=(-1)^{n}L_{n}.\end{aligned}}}$$

The Fibonacci and Lucas numbers can alternately be expressed as the components ⁠${\displaystyle b}$⁠ and ⁠${\displaystyle a}$⁠ when a power of the golden ratio or its conjugate is written in the form ⁠${\displaystyle {\tfrac {1}{2}}a+{\tfrac {1}{2}}b{\sqrt {5}}}$⁠:^[38] $${\displaystyle {\begin{aligned}\varphi ^{n}&={\tfrac {1}{2}}L_{n}+{\tfrac {1}{2}}F_{n}{\sqrt {5}},\\[3mu]{\overline {\varphi }}^{n}&={\tfrac {1}{2}}L_{n}-{\tfrac {1}{2}}F_{n}{\sqrt {5}}.\end{aligned}}}$$

The expression of the Fibonacci numbers in terms of ⁠${\displaystyle \varphi }$⁠ is called Binet's formula:^[39]

$${\displaystyle {\begin{aligned}F_{n}&={\frac {\varphi ^{n}-{\overline {\varphi }}^{n}}{\varphi -{\overline {\varphi }}}}={\frac {\varphi ^{n}-{\overline {\varphi }}^{n}}{\sqrt {5}}}={\frac {\mathrm {Tr} {\bigl (}\varphi ^{n}{\sqrt {5}}~\!{\bigr )}}{5}},\\[5mu]L_{n}&={\frac {\varphi ^{n}+{\overline {\varphi }}^{n}}{\varphi +{\overline {\varphi }}}}=\varphi ^{n}+{\overline {\varphi }}^{n}=\mathrm {Tr} {\left(\varphi ^{n}\right)}.\end{aligned}}}$$

The powers of ⁠${\displaystyle \varphi }$⁠ or ⁠${\displaystyle {\overline {\varphi }}}$⁠, when written in the form ⁠${\displaystyle a+b\varphi }$⁠, can be expressed in terms of just Fibonacci numbers,^[20] $${\displaystyle {\begin{aligned}\varphi ^{n}&=F_{n-1}+F_{n}\varphi ,\\[3mu]{\overline {\varphi }}^{n}&=F_{n-1}+F_{n}{\overline {\varphi }}=F_{n+1}-F_{n}\varphi .\end{aligned}}}$$ and powers of ⁠${\displaystyle \varphi }$⁠ or ⁠${\displaystyle {\overline {\varphi }}}$⁠ times ⁠${\displaystyle {\sqrt {5}}}$⁠ can be expressed in terms of just Lucas numbers, $${\displaystyle {\begin{aligned}\varphi ^{n}{\sqrt {5}}&=L_{n-1}+L_{n}\varphi ,\\[3mu]{\overline {\varphi }}^{n}{\sqrt {5}}&=-L_{n-1}-L_{n}{\overline {\varphi }}=-L_{n+1}+L_{n}\varphi .\end{aligned}}}$$

The numbers ⁠${\displaystyle \textstyle \varphi ^{n}}$⁠ and ⁠${\displaystyle \textstyle {\overline {\varphi }}^{n}}$⁠ are the roots of the quadratic polynomial ⁠${\displaystyle \textstyle x^{2}-L_{n}x+(-1)^{n}}$⁠. This is the minimal polynomial for ⁠${\displaystyle \textstyle \varphi ^{n}}$⁠ for any non-zero integer ⁠${\displaystyle n}$⁠.^[40] The quadratic polynomial ⁠${\displaystyle \textstyle x^{2}-5F_{n}x+(-1)^{n+1}5}$⁠ is the minimal polynomial for ⁠${\displaystyle \textstyle \varphi ^{n}{\sqrt {5}}}$⁠.^[41]

In the limit, consecutive Fibonacci or Lucas numbers approach a ratio of ⁠${\displaystyle \varphi }$⁠, and the ratio of Lucas to Fibonacci numbers approaches ⁠${\displaystyle {\sqrt {5}}}$⁠:^[4] $${\displaystyle {\begin{aligned}\lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}&=\lim _{n\to \infty }{\frac {L_{n+1}}{L_{n}}}=\varphi ,&\lim _{n\to \infty }{\frac {L_{n}}{F_{n}}}&={\sqrt {5}}.\end{aligned}}}$$

Theorems about the Fibonacci numbers – for example, divisibility properties such as if ⁠${\displaystyle n}$⁠ divides ⁠${\displaystyle m}$⁠ then ⁠${\displaystyle F_{n}}$⁠ divides ⁠${\displaystyle F_{m}}$⁠ – can be conveniently proven using ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠.^[42]

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 all five sum to zero. However, ⁠${\displaystyle \mathbb {Q} (\zeta )}$⁠ is only of degree two over ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠, $${\displaystyle {\begin{aligned}x^{5}-1&=(x-1){\bigl (}x^{4}+x^{3}+x^{2}+x+1{\bigr )}\\[2mu]&=(x-1){\bigl (}x^{2}+{\overline {\varphi }}x+1{\bigr )}{\bigl (}x^{2}+\varphi x+1{\bigr )}\\[2mu]&=(x-1){\bigl (}x-\zeta {\bigr )}{\bigl (}x-\zeta ^{4}{\bigr )}{\bigl (}x-\zeta ^{2}{\bigr )}{\bigl (}x-\zeta ^{3}{\bigr )},\end{aligned}}}$$ where the conjugate ⁠${\displaystyle {\overline {\varphi }}=1-\varphi }$⁠. 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 ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠: $${\displaystyle {\begin{aligned}&a_{0}+a_{1}\zeta +a_{2}\zeta ^{2}+a_{3}\zeta ^{3}+a_{4}\zeta ^{4}\\[3mu]&\qquad ={\bigl (}a_{0}-a_{2}+{\overline {\varphi }}a_{3}-{\overline {\varphi }}a_{4}{\bigr )}+{\bigl (}a_{1}-{\overline {\varphi }}a_{2}+{\overline {\varphi }}a_{3}-a_{4}{\bigr )}\zeta .\end{aligned}}}$$

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 ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠.^[43]

Golden integers are involved in the trigonometric study of fivefold symmetries. By the quadratic formula, $${\displaystyle {\begin{alignedat}{3}\zeta &={\tfrac {1}{2}}{\bigl (}{-{\overline {\varphi }}}+{\textstyle {\sqrt {\,{\overline {\varphi }}{}^{2}-4}}}~\!{\bigr )}&{}=-{\tfrac {1}{2}}{\overline {\varphi }}+{\tfrac {1}{2}}{\sqrt {-2-\varphi }},\\[6mu]\zeta ^{2}&={\tfrac {1}{2}}{\bigl (}{-\varphi }+{\textstyle {\sqrt {\varphi {}^{2}-4}}}~\!{\bigr )}&{}=-{\tfrac {1}{2}}\varphi +{\textstyle {\tfrac {1}{2}}{\sqrt {-2-{\overline {\varphi }}}}}.\end{alignedat}}}$$

Angles of ⁠${\displaystyle {\tfrac {2}{5}}\pi }$⁠ and ⁠${\displaystyle {\tfrac {4}{5}}\pi }$⁠ thus have golden rational cosines but their sines are the square roots of golden rational numbers.^[44] $${\displaystyle {\begin{aligned}\cos {\tfrac {2}{5}}\pi &=-{\tfrac {1}{2}}{\overline {\varphi }},&\sin {\tfrac {2}{5}}\pi &={\tfrac {1}{2}}{\sqrt {2+\varphi }},\\[6mu]\cos {\tfrac {4}{5}}\pi &=-{\tfrac {1}{2}}\varphi ,&\sin {\tfrac {4}{5}}\pi &={\tfrac {1}{2}}{\textstyle {\sqrt {2+{\overline {\varphi }}}}}.\end{aligned}}}$$ The numbers ⁠${\displaystyle 2+\varphi =\varphi {\sqrt {5}}}$⁠ and ⁠${\displaystyle 2+{\overline {\varphi }}=3-\varphi =\varphi ^{-1}{\sqrt {5}}}$⁠ are conjugates with norm ⁠${\displaystyle 5}$⁠. These are the squared Euclidean lengths of the diagonal and side, respectively, of a regular pentagon with unit circumradius.

Three-dimensional space

A regular icosahedron with edge length ⁠${\displaystyle 2}$⁠ can be oriented so that the Cartesian coordinates of its vertices are^[45] $${\displaystyle \left(0,\pm 1,\pm \varphi \right),\left(\pm 1,\pm \varphi ,0\right),\left(\pm \varphi ,0,\pm 1\right).}$$

Four-dimensional space

The 600-cell is a regular 4-polytope with 120 vertices, 720 edges, 1200 triangular faces, and 600 tetrahedral cells. It has kaleidoscopic symmetry ⁠${\displaystyle [5,3,3]}$⁠ generated by four mirrors which can be conveniently oriented as ⁠${\displaystyle {\sqrt {5}}x_{1}=x_{2}+x_{3}+x_{4}}$⁠, ⁠${\displaystyle x_{1}=x_{2}}$⁠, ⁠${\displaystyle x_{2}=x_{3}}$⁠, and ⁠${\displaystyle x_{3}=x_{4}}$⁠. Then the 120 vertices have golden-integer coordinates: arbitrary permutations of ⁠${\displaystyle \textstyle (\varphi ,\varphi ,\varphi ,\varphi ^{-2})}$⁠ and ⁠${\displaystyle \textstyle (\varphi ^{-1},\varphi ^{-1},\varphi ^{-1},\varphi ^{2})}$⁠ with an even number of minus signs, ⁠${\displaystyle (1,1,1,{\sqrt {5}})}$⁠ with an odd number of minus signs, and ⁠${\displaystyle (\pm 2,\pm 2,0,0)}$⁠.^[46][47]

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.^[48] The icosians of unit norm are the vertices of a 600-cell.^[47]

Quasiperiodicity

The Fibonacci chain, a one-dimensional quasicrystal, constructed by the cut-and-project method

Golden integers are used in studying quasicrystals.^[49]

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.^[50]

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.^[51] They can, in fact, be defined over the golden field.^[52]

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.^[53]