Golden field

In mathematics, ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠, sometimes called the golden field, 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 ⁠$a+b{\sqrt {5}}$⁠, where ⁠$a$⁠ and ⁠$b$⁠ are both rational numbers. As a field, ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠ supports the same basic arithmetical operations as the rational numbers. The name comes from the golden ratio ⁠$\varphi$⁠, which is the fundamental unit of ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠, and which satisfies the equation ⁠$\textstyle \varphi ^{2}=\varphi +1$⁠.

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 ⁠$a+b{\sqrt {5}}$⁠ where ⁠$a$⁠ and ⁠$b$⁠ are uniquely determined rational numbers, or in the form ⁠${\bigl (}a+b{\sqrt {5}}~\!{\bigr )}{\big /}c$⁠ where ⁠$a$⁠, ⁠$b$⁠, and ⁠$c$⁠ are integers, which can be uniquely reduced to lowest terms, and where ⁠${\sqrt {5}}=2.236\ldots$⁠ is the square root of 5. It is sometimes more convenient instead to use the form ⁠$a+b\varphi$⁠ where ⁠$a$⁠ and ⁠$b$⁠ are rational or the form ⁠$(a+b\varphi )/c$⁠ where ⁠$a$⁠, ⁠$b$⁠, and ⁠$c$⁠ are integers, and where $\varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}={}\!$$1.618\ldots$ is the golden ratio.

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

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

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

To find the reciprocal of a number $\alpha$, rationalize the denominator: $1/\alpha ={}$${\overline {\alpha }}/\alpha {\overline {\alpha }}={}$${\overline {\alpha }}/\mathrm {N} (\alpha )$, where ⁠${\overline {\alpha }}$⁠ is the algebraic conjugate and ⁠$\mathrm {N} (\alpha )$⁠ is the field norm, as defined below. Explicitly: $${\begin{aligned}{\frac {1}{a+b{\sqrt {5}}}}&={\frac {1}{a+b{\sqrt {5}}}}\cdot {\frac {a-b{\sqrt {5}}}{a-b{\sqrt {5}}}}={\frac {a}{a^{2}-5b^{2}}}-{\frac {b}{a^{2}-5b^{2}}}{\sqrt {5}},\\[3mu]{\frac {1}{a+b\varphi }}&={\frac {1}{a+b\varphi }}\cdot {\frac {a+b-b\varphi }{a+b-b\varphi }}={\frac {a+b}{a^{2}+ab-b^{2}}}-{\frac {b}{a^{2}+ab-b^{2}}}\varphi .\end{aligned}}$$

To divide two numbers, multiply the first by second's reciprocal: $${\begin{aligned}{\frac {a_{1}+b_{1}{\sqrt {5}}}{a_{2}+b_{2}{\sqrt {5}}}}&={\frac {a_{1}a_{2}-5b_{1}b_{2}}{a_{2}^{2}-5b_{2}^{2}}}+{\frac {-a_{1}b_{2}+b_{1}a_{2}}{a_{2}^{2}-5b_{2}^{2}}}{\sqrt {5}},\\[6mu]{\frac {a_{1}+b_{1}\varphi }{a_{2}+b_{2}\varphi }}&={\frac {a_{1}a_{2}+a_{1}b_{2}-b_{1}b_{2}}{a_{2}^{2}+a_{2}b_{2}-b_{2}^{2}}}+{\frac {-a_{1}b_{2}+b_{1}a_{2}}{a_{2}^{2}+a_{2}b_{2}-b_{2}^{2}}}\varphi .\end{aligned}}$$

Conjugation and norm

The numbers ⁠${\sqrt {5}}$⁠ and ⁠$-{\sqrt {5}}$⁠ each solve the equation ⁠$\textstyle x^{2}=5$⁠. Each number ⁠$\alpha =a+b{\sqrt {5}}$⁠ in ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠ has an algebraic conjugate ⁠${\overline {\alpha }}$⁠ found by swapping these two square roots of 5, i.e., by changing the sign of ⁠$b$⁠. The conjugate of ⁠$\varphi$⁠ is ${\overline {\varphi }}={\tfrac {1}{2}}{\bigl (}1-{\sqrt {5}}~\!{\bigr )}={}$$\textstyle -\varphi ^{-1}={}$$1-\varphi$. A rational number is its own conjugate. In general, the conjugate is: $${\begin{aligned}{\overline {a+b{\sqrt {5}}}}&=a-b{\sqrt {5}},\\[3mu]{\overline {a+b\varphi }}&=a+b{\overline {\varphi }}=(a+b)-b\varphi .\end{aligned}}$$ Conjugation in ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠ is an involution, ⁠${\overline {({\overline {\alpha }})}}=\alpha$⁠, and it preserves the structure of arithmetic: ⁠${\overline {\alpha _{1}+\alpha _{2}}}={\overline {\alpha }}_{1}+{\overline {\alpha }}_{2}$⁠; ⁠${\overline {\alpha _{1}\alpha _{2}}}={\overline {\alpha }}_{1}{\overline {\alpha }}_{2}$⁠; and ⁠${\overline {\alpha _{1}/\alpha _{2}}}={\overline {\alpha }}_{1}/\,{\overline {\alpha }}_{2}$⁠. Conjugation is the only ring homomorphism (function preserving the structure of addition and multiplication) from ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠ to itself, other than the identity function.

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). The trace of ⁠$\alpha$⁠ in ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠ is ⁠$\mathrm {Tr} (\alpha )=\alpha +{\overline {\alpha }}$⁠: $${\begin{aligned}\mathrm {Tr} {\bigl (}a+b{\sqrt {5}}~\!{\bigr )}&={\bigl (}a+b{\sqrt {5}}~\!{\bigr )}+{\bigl (}a-b{\sqrt {5}}~\!{\bigr )}=2a,\\[3mu]\mathrm {Tr} (a+b\varphi )&=(a+b\varphi )+(a+b-b\varphi )=2a+b.\end{aligned}}$$ This is always an (ordinary) rational number.

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

The norm preserves the structure of multiplication, as expected for a concept of magnitude. The norm of a product is the product of norms, ⁠$\operatorname {N} (\alpha _{1}\alpha _{2})=\mathrm {N} (\alpha _{1})~\!\mathrm {N} (\alpha _{2})$⁠; and the norm of a quotient is the quotient of the norms, ⁠$\mathrm {N} (\alpha _{1}/\alpha _{2})=\mathrm {N} (\alpha _{1}){\big /}~\!\mathrm {N} (\alpha _{2})$⁠. A number and its conjugate have the same norm, ⁠$\mathrm {N} (\alpha )=\mathrm {N} ({\overline {\alpha }})$⁠;

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

In Galois theory, the golden field can be considered more abstractly as the set of all numbers ⁠$a+bu$⁠, where ⁠$a$⁠ and ⁠$b$⁠ are both rational, and all that is known of ⁠$u$⁠ is that it satisfies the equation ⁠$\textstyle u^{2}=5$⁠. There are two ways to embed this set in the real numbers: by mapping ⁠$u$⁠ to the positive square root ⁠${\sqrt {5}}$⁠ or alternatively by mapping ⁠$u$⁠ to the negative square root ⁠$-{\sqrt {5}}$⁠. 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.

Golden integers

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

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

⁠$0$⁠, ⁠$1$⁠, ⁠$4$⁠, ⁠$5$⁠, ⁠$9$⁠, ⁠$11$⁠, ⁠$16$⁠, ⁠$19$⁠, ⁠$20$⁠, ⁠$25$⁠, ⁠$29$⁠, . . . .

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

A unit is an algebraic integer whose multiplicative inverse is also an algebraic integer, which happens when its norm is ⁠$\pm 1$⁠. Thus the units are all numbers of the form ⁠$a+b\varphi$⁠ whose integer coefficients ⁠$a$⁠ and ⁠$b$⁠ solve the Diophantine equation ⁠$\textstyle a^{2}+ab-b^{2}=\pm 1$⁠. If the unit is instead written in the form ⁠${\tfrac {1}{2}}a+{\tfrac {1}{2}}b{\sqrt {5}}$⁠, the coefficients solve a related Diophantine equation, the generalized Pell's equation ⁠$\textstyle a^{2}-5b^{2}=\pm 4$⁠. The fundamental unit is the golden ratio ⁠$\varphi ={\tfrac {1}{2}}+{\tfrac {1}{2}}{\sqrt {5}}$⁠ and the other units are its positive and negative powers, ⁠$\pm \varphi ^{n}$⁠, for any integer ⁠$n$⁠. Some powers of ⁠$\varphi$⁠ are:

⁠${\boldsymbol {n}}$⁠	⁠$\ldots$⁠	⁠$-2$⁠	⁠$-1$⁠	⁠$0$⁠	⁠$1$⁠	⁠$2$⁠	⁠$3$⁠	⁠$4$⁠	⁠$5$⁠	⁠$\ldots$⁠	⁠$n$⁠
⁠${\boldsymbol {\varphi ^{n}}}$⁠	⁠$\ldots$⁠	⁠$2-\varphi$⁠	⁠$-1+\varphi$⁠	⁠$1$⁠	⁠$\varphi$⁠	⁠$1+\varphi$⁠	⁠$1+2\varphi$⁠	⁠$2+3\varphi$⁠	⁠$3+5\varphi$⁠	⁠$\ldots$⁠	⁠$F_{n-1}+F_{n}\varphi$⁠

In general ⁠$\textstyle \varphi ^{n}=F_{n-1}+F_{n}\varphi$⁠, where ⁠$F_{n}$⁠ is the ⁠$n$⁠th Fibonacci number. The units form a group under multiplication, which can be decomposed as the direct product ⁠$\mathbb {Z} _{2}\times \mathbb {Z}$⁠ of a cyclic group of order 2 and an infinite cyclic group, respectively generated by ⁠$-1$⁠ and ⁠$\varphi$⁠.

Two golden integers are associates if their quotient in ⁠$\mathbb {Q} (\varphi )$⁠ is a unit; that is, two golden integers ⁠$\alpha _{1}$⁠ and ⁠$\alpha _{2}$⁠ are associates if ⁠$\alpha _{2}=\pm \varphi ^{n}\alpha _{1}$⁠ for some integer ⁠$n$⁠. Associateness is an equivalence relation. Associates have the same norm, up to sign: $|\mathrm {N} (\alpha _{1})|=|\mathrm {N} (\alpha _{2})|$. However, not all elements whose norm has the same absolute value are associates; in particular, any golden prime and its conjugate have the same norm, but are associates if and only if they are associated either with ⁠${\sqrt {5}}$⁠ or with an ordinary prime.

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

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

The ring ⁠$\mathbb {Z} [\varphi ]$⁠ 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. This makes ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠ one of the 21 quadratic fields that are norm-Euclidean.

Like all Euclidean domains, the ring ⁠$\mathbb {Z} [\varphi ]$⁠ 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 ⁠$\mathbb {Z} [\varphi ]$⁠ 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).

In the table below, positive golden integers have been arranged into rows, with one representative chosen for each class of associates (here the representative is the positive element ⁠$\alpha$⁠ in the class for which $\alpha +|{\overline {\alpha }}|$ is a minimum).

Rep.	Norm	Trace	Factorization	Positive associates
⁠${\phantom {-}}0$⁠	⁠${\phantom {-}}0$⁠	⁠${\phantom {-}}0$⁠	zero
⁠${\phantom {-}}1$⁠	⁠${\phantom {-}}1$⁠	⁠${\phantom {-}}2$⁠	unit	⁠$\ldots$⁠, ⁠$-3+2\varphi$⁠, ⁠$2-\varphi$⁠, ⁠$-1+\varphi$⁠; ⁠$\varphi$⁠, ⁠$1+\varphi$⁠, ⁠$1+2\varphi$⁠, ⁠$\ldots$⁠
⁠${\phantom {-}}2$⁠	⁠${\phantom {-}}4$⁠	⁠${\phantom {-}}4$⁠	prime	⁠$\ldots$⁠, ⁠$4-2\varphi$⁠, ⁠$-2+2\varphi$⁠; ⁠$2\varphi$⁠, ⁠$2+2\varphi$⁠, ⁠$\ldots$⁠
⁠$-1+2\varphi$⁠	⁠$-5$⁠	⁠${\phantom {-}}0$⁠	prime	⁠$\ldots$⁠, ⁠$-4+3\varphi$⁠, ⁠$3-\varphi$⁠; ⁠$2+\varphi$⁠, ⁠$1+3\varphi$⁠, ⁠$\ldots$⁠
⁠${\phantom {-}}3$⁠	⁠${\phantom {-}}9$⁠	⁠${\phantom {-}}6$⁠	prime	⁠$\ldots$⁠, ⁠$6-3\varphi$⁠, ⁠$-3+3\varphi$⁠; ⁠$3\varphi$⁠, ⁠$3+3\varphi$⁠, ⁠$\ldots$⁠
⁠$-1+3\varphi$⁠	⁠$-11$⁠	⁠${\phantom {-}}1$⁠	prime	⁠$\ldots$⁠, ⁠$-5+4\varphi$⁠, ⁠$4-\varphi$⁠; ⁠$3+2\varphi$⁠, ⁠$2-5\varphi$⁠, ⁠$\ldots$⁠
⁠$-2+3\varphi$⁠	⁠$-11$⁠	⁠$-1$⁠	prime	⁠$\ldots$⁠, ⁠$-7+5\varphi$⁠, ⁠$5-2\varphi$⁠; ⁠$3+\varphi$⁠, ⁠$1+4\varphi$⁠ ⁠$\ldots$⁠
⁠${\phantom {-}}4$⁠	⁠${\phantom {-}}16$⁠	⁠${\phantom {-}}8$⁠	⁠$2^{2}$⁠	⁠$\ldots$⁠, ⁠$8-4\varphi$⁠, ⁠$-4+4\varphi$⁠; ⁠$4\varphi$⁠, ⁠$4+4\varphi$⁠, ⁠$\ldots$⁠
⁠$-1+4\varphi$⁠	⁠$-19$⁠	⁠${\phantom {-}}2$⁠	prime	⁠$\ldots$⁠, ⁠$-6+5\varphi$⁠, ⁠$5-\varphi$⁠; ⁠$4+3\varphi$⁠, ⁠$3+7\varphi$⁠, ⁠$\ldots$⁠
⁠$-3+4\varphi$⁠	⁠$-19$⁠	⁠$-2$⁠	prime	⁠$\ldots$⁠, ⁠$-10+7\varphi$⁠, ⁠$7-3\varphi$⁠; ⁠$4+\varphi$⁠, ⁠$1+5\varphi$⁠, ⁠$\ldots$⁠
⁠$-2+4\varphi$⁠	⁠$-20$⁠	⁠${\phantom {-}}0$⁠	⁠$2(-1+2\varphi )$⁠	⁠$\ldots$⁠, ⁠$-8+6\varphi$⁠, ⁠$6-2\varphi$⁠; ⁠$4+2\varphi$⁠, ⁠$2+6\varphi$⁠, ⁠$\ldots$⁠
⁠${\phantom {-}}5$⁠	⁠${\phantom {-}}25$⁠	⁠${\phantom {-}}10$⁠	⁠$(-1+2\varphi )^{2}$⁠	⁠$\ldots$⁠, ⁠$10-5\varphi$⁠, ⁠$-5+5\varphi$⁠; ⁠$5\varphi$⁠, ⁠$5+5\varphi$⁠, ⁠$\ldots$⁠
⁠${\phantom {-}}5+\varphi$⁠	⁠${\phantom {-}}29$⁠	⁠${\phantom {-}}11$⁠	prime	⁠$\ldots$⁠, ⁠$9-4\varphi$⁠, ⁠$-4+5\varphi$⁠; ⁠$1+6\varphi$⁠, ⁠$6+7\varphi$⁠, ⁠$\ldots$⁠
⁠${\phantom {-}}6-\varphi$⁠	⁠${\phantom {-}}29$⁠	⁠${\phantom {-}}11$⁠	prime	⁠$\ldots$⁠, ⁠$13-7\varphi$⁠, ⁠$-7+6\varphi$⁠; ⁠$-1+5\varphi$⁠, ⁠$5+4\varphi$⁠, ⁠$\ldots$⁠
⁠$-2+5\varphi$⁠	⁠$-31$⁠	⁠${\phantom {-}}1$⁠	prime	⁠$\ldots$⁠, ⁠$-9+7\varphi$⁠, ⁠$7-2\varphi$⁠; ⁠$5+3\varphi$⁠, ⁠$3+8\varphi$⁠, ⁠$\ldots$⁠
⁠$-3+5\varphi$⁠	⁠$-31$⁠	⁠$-1$⁠	prime	⁠$\ldots$⁠, ⁠$-11+8\varphi$⁠, ⁠$8-3\varphi$⁠; ⁠$5+2\varphi$⁠, ⁠$2+7\varphi$⁠, ⁠$\ldots$⁠
⁠${\phantom {-}}6$⁠	⁠${\phantom {-}}36$⁠	⁠${\phantom {-}}12$⁠	⁠$2\cdot 3$⁠	⁠$\ldots$⁠, ⁠$12-6\varphi$⁠, ⁠$-6+6\varphi$⁠; ⁠$6\varphi$⁠, ⁠$6+6\varphi$⁠, ⁠$\ldots$⁠
⁠${\phantom {-}}6+\varphi$⁠	⁠${\phantom {-}}41$⁠	⁠${\phantom {-}}13$⁠	prime	⁠$\ldots$⁠, ⁠$11-5\varphi$⁠, ⁠$-5+6\varphi$⁠; ⁠$1+7\varphi$⁠, ⁠$7+8\varphi$⁠, ⁠$\ldots$⁠
⁠${\phantom {-}}7-\varphi$⁠	⁠${\phantom {-}}41$⁠	⁠${\phantom {-}}13$⁠	prime	⁠$\ldots$⁠, ⁠$15-8\varphi$⁠, ⁠$-8+7\varphi$⁠; ⁠$-1+6\varphi$⁠, ⁠$6+5\varphi$⁠, ⁠$\ldots$⁠
⁠$-2+6\varphi$⁠	⁠$-44$⁠	⁠${\phantom {-}}2$⁠	⁠$2(-1+3\varphi )$⁠	⁠$\ldots$⁠, ⁠$-10+8\varphi$⁠, ⁠$8-2\varphi$⁠; ⁠$6+4\varphi$⁠, ⁠$4+10\varphi$⁠, ⁠$\ldots$⁠
⁠$-4+6\varphi$⁠	⁠$-44$⁠	⁠$-2$⁠	⁠$2(-2+3\varphi )$⁠	⁠$\ldots$⁠, ⁠$-14+10\varphi$⁠, ⁠$10-4\varphi$⁠; ⁠$6+2\varphi$⁠, ⁠$2+8\varphi$⁠, ⁠$\ldots$⁠
⁠$-3+6\varphi$⁠	⁠$-45$⁠	⁠$0$⁠	⁠$3(-1+2\varphi )$⁠	⁠$\ldots$⁠, ⁠$-12+9\varphi$⁠, ⁠$9-3\varphi$⁠; ⁠$6+3\varphi$⁠, ⁠$3+9\varphi$⁠, ⁠$\ldots$⁠
⁠${\phantom {-}}7$⁠	⁠$49$⁠	⁠$14$⁠	prime	⁠$\ldots$⁠, ⁠$14-7\varphi$⁠, ⁠$-7+7\varphi$⁠; ⁠$7\varphi$⁠, ⁠$7+7\varphi$⁠, ⁠$\ldots$⁠
⁠${\phantom {-}}7+\varphi$⁠	⁠${\phantom {-}}55$⁠	⁠$15$⁠	⁠$(-1+2\varphi )(-1+3\varphi )$⁠	⁠$\ldots$⁠, ⁠$13-6\varphi$⁠, ⁠$-6+7\varphi$⁠; ⁠$1+8\varphi$⁠, ⁠$8+9\varphi$⁠, ⁠$\ldots$⁠
⁠${\phantom {-}}8-\varphi$⁠	⁠${\phantom {-}}55$⁠	⁠$15$⁠	⁠$(-1+2\varphi )(-2+3\varphi )$⁠	⁠$\ldots$⁠, ⁠$17-9\varphi$⁠, ⁠$-9+8\varphi$⁠; ⁠$-1+7\varphi$⁠, ⁠$7+6\varphi$⁠, ⁠$\ldots$⁠
⁠$-2+7\varphi$⁠	⁠$-59$⁠	⁠$3$⁠	prime	⁠$\ldots$⁠, ⁠$-11+9\varphi$⁠, ⁠$9-2\varphi$⁠; ⁠$7+5\varphi$⁠, ⁠$5+12\varphi$⁠, ⁠$\ldots$⁠
⁠$-5+7\varphi$⁠	⁠$-59$⁠	⁠$-3$⁠	prime	⁠$\ldots$⁠, ⁠$-17+12\varphi$⁠, ⁠$12-5\varphi$⁠; ⁠$7+2\varphi$⁠, ⁠$2+9\varphi$⁠, ⁠$\ldots$⁠
⁠$\quad \!\vdots$⁠ Show/hide more rows

Matrix representation

⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠ is a two-dimensional vector space over ⁠$\mathbb {Q}$⁠, and multiplication by any element of ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠ is a linear transformation of that vector space. Given an ordered basis of ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠, each number in ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠ can be associated to the corresponding transformation matrix in that basis. This defines a field isomorphism (a structure-preserving bijective map) from ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠ to the space of ⁠$2\times 2$⁠ square matrices with rational entries spanned by the identity matrix ⁠$\mathbf {I}$⁠, the image of the number ⁠$1$⁠, and a matrix ⁠$\mathbf {\Phi }$⁠, the image of ⁠$\varphi$⁠. Thus arithmetic of numbers in ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠ can be alternately represented by the arithmetic of such matrices. In this context, the number ⁠$\alpha =a+b\varphi$⁠ is represented by the matrix ⁠$\mathbf {A} =a\mathbf {I} +b\mathbf {\Phi }$⁠. A convenient choice of basis for ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠ is ⁠$(1,\varphi )$⁠, in terms of which ⁠$\mathbf {\Phi }$⁠ is a symmetric matrix: $${\begin{aligned}&\mathbf {I} ={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\quad \mathbf {\Phi } ={\begin{bmatrix}0&1\\1&1\end{bmatrix}},\quad \mathbf {A} =a\mathbf {I} +b\mathbf {\Phi } ={\begin{bmatrix}a&b\\b&a+b\end{bmatrix}}.\quad \end{aligned}}$$

The adjugate matrix ⁠${\overline {\mathbf {\Phi } }}=\mathbf {I} -\mathbf {\Phi }$⁠ represents the algebraic conjugate ⁠${\overline {\varphi }}=1-\varphi$⁠, the matrix ⁠$\mathbf {R} =-\mathbf {I} +2\mathbf {\Phi }$⁠ (satisfying ⁠$\textstyle \mathbf {R} ^{2}=5\mathbf {I}$⁠) represents ⁠${\sqrt {5}}$⁠, and the adjugate of an arbitrary element ⁠$\mathbf {A}$⁠, which we will denote ⁠${\overline {\mathbf {A} }}=a\mathbf {I} +b{\overline {\mathbf {\Phi } }}$⁠, represents the number ⁠${\overline {\alpha }}=a+b{\overline {\varphi }}$⁠: $${\begin{aligned}&\mathbf {R} ={\begin{bmatrix}-1&2\\2&1\end{bmatrix}},\quad {\overline {\mathbf {\Phi } }}={\begin{bmatrix}~1&-1\\-1&\,0\end{bmatrix}},\quad {\overline {\mathbf {A} }}=a\mathbf {I} +b{\overline {\mathbf {\Phi } }}={\begin{bmatrix}a+b&-b~\\-b&~a\end{bmatrix}}.\end{aligned}}$$

Every matrix ⁠$\mathbf {A} =a\mathbf {I} +b\mathbf {\Phi }$⁠, except for the zero matrix, is invertible, and its inverse ⁠$\textstyle \mathbf {A} ^{-1}={\bigl (}1/\mathrm {det} (\mathbf {A} ){\bigr )}{\overline {\mathbf {A} }}$⁠ represents the multiplicative inverse ⁠$\textstyle \alpha ^{-1}={\overline {\alpha }}/\mathrm {N} (\alpha )$⁠ in ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠.

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

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

Every matrix of the form ⁠$a\mathbf {I} +b\mathbf {\Phi }$⁠ has eigenvectors which point along the directions ⁠$\textstyle {\begin{bmatrix}1&\varphi \end{bmatrix}}{}^{\top }$⁠ and ⁠$\textstyle {\begin{bmatrix}1&{\overline {\varphi }}\end{bmatrix}}{}^{\top }\!$⁠. When numbers in ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠ are plotted, as above, in a coordinate system where their values as real numbers are the horizontal axis and the values of their conjugates are the vertical axis, the eigenvectors point along those two axes. (Zero is the only number ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠ directly on either axis.) The matrices ⁠$\mathbf {\Phi } ^{2n}$⁠ for integer ⁠$n$⁠, representing units, and more generally any matrices with ⁠$a+b\varphi >0$⁠ and determinant ⁠$1$⁠, are squeeze mappings, which stretch the plane along one axis and squish it along the other, fixing hyperbolas of constant norm. The matrices ⁠$\mathbf {\Phi } ^{2n+1}$⁠ and more generally matrices with ⁠$a+b\varphi >0$⁠ and determinant ⁠$-1$⁠, are the composition of a squeeze mapping and a vertical reflection. The negative identity matrix ⁠$-\mathbf {I}$⁠ is a point reflection across the origin. In general any other matrix ⁠$a\mathbf {I} +b\mathbf {\Phi }$⁠ can be decomposed as the product of a squeeze mapping, possibly a reflection, and a uniform scaling by the square root of the absolute value of its determinant.

Other properties

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

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 $\varphi$.

Fibonacci numbers

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

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

The sequences ⁠$F_{n}$⁠ and ⁠$L_{n}$⁠ respectively begin:

⁠${\boldsymbol {n}}$⁠	⁠${\phantom {0}}0$⁠	⁠${\phantom {0}}1$⁠	⁠${\phantom {0}}2$⁠	⁠${\phantom {0}}3$⁠	⁠${\phantom {0}}4$⁠	⁠${\phantom {0}}5$⁠	⁠${\phantom {0}}6$⁠	⁠${\phantom {0}}7$⁠	⁠${\phantom {0}}8$⁠	⁠${\phantom {0}}9$⁠	⁠$10$⁠	⁠$11$⁠	⁠$12$⁠	⁠$\ldots$⁠
⁠${\boldsymbol {F_{n}}}$⁠	⁠$0$⁠	⁠$1$⁠	⁠$1$⁠	⁠$2$⁠	⁠$3$⁠	⁠$5$⁠	⁠$8$⁠	⁠$13$⁠	⁠$21$⁠	⁠$34$⁠	⁠$55$⁠	⁠$89$⁠	⁠$144$⁠	⁠$\ldots$⁠
⁠${\boldsymbol {L_{n}}}$⁠	⁠$2$⁠	⁠$1$⁠	⁠$3$⁠	⁠$4$⁠	⁠$7$⁠	⁠$11$⁠	⁠$18$⁠	⁠$29$⁠	⁠$47$⁠	⁠$76$⁠	⁠$123$⁠	⁠$199$⁠	⁠$322$⁠	⁠$\ldots$⁠

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

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

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

The expression of the Fibonacci numbers in terms of ⁠$\varphi$⁠ is called Binet's formula:

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

The powers of ⁠$\varphi$⁠ or ⁠${\overline {\varphi }}$⁠, when written in the form ⁠$a+b\varphi$⁠, can be expressed in terms of just Fibonacci numbers, $${\begin{aligned}\varphi ^{n}&=F_{n-1}+F_{n}\varphi ,\\[3mu]{\overline {\varphi }}^{n}&=F_{n-1}+F_{n}{\overline {\varphi }}=F_{n+1}-F_{n}\varphi .\end{aligned}}$$ Powers of ⁠$\varphi$⁠ or ⁠${\overline {\varphi }}$⁠ times ⁠${\sqrt {5}}$⁠ can be expressed in terms of just Lucas numbers, $${\begin{aligned}\varphi ^{n}{\sqrt {5}}&=L_{n-1}+L_{n}\varphi ,\\[3mu]{\overline {\varphi }}^{n}{\sqrt {5}}&=-L_{n-1}-L_{n}{\overline {\varphi }}=-L_{n+1}+L_{n}\varphi .\end{aligned}}$$ Statements about golden integers can be recast as statements about the Fibonacci or Lucas numbers; for example, that every power of ⁠$\varphi$⁠ is a unit of ⁠$\mathbb {Z} [\varphi ]$⁠, ⁠$\textstyle \mathrm {N} (\varphi ^{n})={\mathrm {N} (\varphi )}^{n}={(-1)}^{n}$⁠, when expanded, becomes Cassini's identity, and likewise ⁠$\textstyle \mathrm {N} (\varphi ^{n}{\sqrt {5}})={\mathrm {N} (\varphi )}^{n}\mathrm {N} {\bigl (}{\sqrt {5}}{\bigr )}={(-1)}^{n}5$⁠ becomes the analogous identity about Lucas numbers, $${\begin{aligned}(F_{n-1}+F_{n}\varphi )(F_{n+1}-F_{n}\varphi )&=F_{n-1}F_{n+1}-F_{n}^{2}=(-1)^{n},\\[3mu](L_{n-1}+L_{n}\varphi )(-L_{n+1}+L_{n}\varphi )&=L_{n}^{2}-L_{n-1}L_{n+1}=(-1)^{n}5.\end{aligned}}$$

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

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

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

Relation to fivefold symmetry

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

Conversely, ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠ is a subfield of ⁠$\mathbb {Q} (\zeta )$⁠. For any primitive root of unity ⁠$\zeta _{n}$⁠, the maximal real subfield of the cyclotomic field ⁠$\mathbb {Q} (\zeta _{n})$⁠ is the field ⁠$\mathbb {Q} (\zeta _{n}+\zeta _{n}^{-1})$⁠; see Minimal polynomial of ⁠$2\cos(2\pi /n)$⁠. In our case ⁠$n=5$⁠, ⁠$\zeta +\zeta ^{-1}=\varphi ^{-1}$⁠, so the maximal real subfield of ⁠$\mathbb {Q} (\zeta )$⁠ is ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠.

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

Angles of ⁠${\tfrac {2}{5}}\pi$⁠ and ⁠${\tfrac {4}{5}}\pi$⁠ thus have golden rational cosines but their sines are the square roots of golden rational numbers. $${\begin{aligned}\cos {\tfrac {2}{5}}\pi &=-{\tfrac {1}{2}}{\overline {\varphi }},&\sin {\tfrac {2}{5}}\pi &={\tfrac {1}{2}}{\sqrt {2+\varphi }},\\[6mu]\cos {\tfrac {4}{5}}\pi &=-{\tfrac {1}{2}}\varphi ,&\sin {\tfrac {4}{5}}\pi &={\tfrac {1}{2}}{\textstyle {\sqrt {2+{\overline {\varphi }}}}}.\end{aligned}}$$ The numbers ⁠$2+\varphi =\varphi {\sqrt {5}}$⁠ and ⁠$2+{\overline {\varphi }}=3-\varphi =\varphi ^{-1}{\sqrt {5}}$⁠ are conjugates with norm ⁠$5$⁠. 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 ⁠$2$⁠ can be oriented so that the Cartesian coordinates of its vertices are $$\left(0,\pm 1,\pm \varphi \right),\left(\pm 1,\pm \varphi ,0\right),\left(\pm \varphi ,0,\pm 1\right).$$

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 ⁠$[5,3,3]$⁠ generated by four mirrors which can be conveniently oriented as ⁠${\sqrt {5}}x_{1}=x_{2}+x_{3}+x_{4}$⁠, ⁠$x_{1}=x_{2}$⁠, ⁠$x_{2}=x_{3}$⁠, and ⁠$x_{3}=x_{4}$⁠. Then the 120 vertices have golden-integer coordinates: arbitrary permutations of ⁠$\textstyle (\varphi ,\varphi ,\varphi ,\varphi ^{-2})$⁠ and ⁠$\textstyle (\varphi ^{-1},\varphi ^{-1},\varphi ^{-1},\varphi ^{2})$⁠ with an even number of minus signs, ⁠$(1,1,1,{\sqrt {5}})$⁠ with an odd number of minus signs, and ⁠$(\pm 2,\pm 2,0,0)$⁠.

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. The icosians of unit norm are the vertices of a 600-cell.

Quasiperiodicity

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

Golden integers are used in studying quasicrystals.

Other applications

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

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. They can, in fact, be defined over the golden field.

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.

References

• Appleby, Marcus; Bengtsson, Ingemar; Grassl, Markus; Harrison, Michael; McConnell, Gary (2022). "SIC-POVMs from Stark units: Prime dimensions ⁠$\textstyle n^{2}+3$⁠". Journal of Mathematical Physics. 63: 112205. arXiv:2112.05552. doi:10.1063/5.0083520.
• Baez, John (1 Mar 2016). "Clebsch Surface". Visual Insight (Blog). American Mathematical Society. Retrieved 2025-06-22.
• Bengtsson, Ingemar (2017). "The number behind the simplest SIC-POVM". Foundations of Physics. 47: 1031–1041. arXiv:1611.09087. doi:10.1007/s10701-017-0078-3.
• Bernat, Julien (2006). "Continued fractions and numeration in the Fibonacci base". Discrete Mathematics. 306 (22): 2828–2850. doi:10.1016/j.disc.2006.05.020.
• Bicknell, Marjorie; Hoggatt, Verner E. Jr. (1973). A Primer for the Fibonacci Numbers. Fibonacci Association.
• Boukas, Andreas; Feinsilver, Philip; Fellouris, Anargyros (2016). "Probability distributions and orthogonal polynomials associated with the one-parameter Fibonacci group". Communications on Stochastic Analysis. 10 (1): 117–130. doi:10.31390/cosa.10.1.08.
• Bradie, Brian (2002). "Exact Values for the Sine and Cosine of Multiples of ⁠$18^{\circ }$⁠ – A Geometric Approach". Classroom Capsules. The College Mathematics Journal. 33 (4): 318–319. doi:10.1080/07468342.2002.11921960.
• Carlitz, L. (1964). "A Note on Fibonacci Numbers" (PDF). The Fibonacci Quarterly. 2 (1): 15–28. doi:10.1080/00150517.1964.12431525.
• Conway, John H.; Sloane, Neil J. A. (1999). "Algebraic Constructions for Lattices". Sphere Packings, Lattices and Groups (3rd ed.). New York: Springer. pp. 206–244. doi:10.1007/978-1-4757-6568-7_8. ISBN 0-387-98585-9.
• Dembélé, Lassina (2005). "Explicit computations of Hilbert modular forms on ⁠$\mathbb {Q} ({\sqrt {5}}~\!)$⁠" (PDF). Experimental Mathematics. 14 (4): 457–466. doi:10.1080/10586458.2005.10128939. Zbl 1152.11328.
• Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernisha (2020). "The geometry of H₄ polytopes". Advances in Geometry. 20 (3): 433–444. arXiv:1912.06156. doi:10.1515/advgeom-2020-0005.
• Dickson, Leonard E. (1923). Algebras and their Arithmetics. University of Chicago Press.
• Dimitrov, Vassil S.; Cosklev, T. V.; Bonevsky, B. (1995). "Number theoretic transforms over the golden section quadratic field". IEEE Transactions on Signal Processing. 43 (8): 1790–1797. doi:10.1109/78.403338.
• Dirichlet, Gustav Lejeune (1828). "Mémoire sur l'impossibilité de quelques équations indéterminées du cinquième degré". Journal für die reine und angewandte Mathematik. 3: 354–375. doi:10.1515/crll.1828.3.354.
• Dodd, Fred W. (1983). Number theory in the quadratic field with golden section unit. Passaic, NJ: Polygonal Publishing House. ISBN 0-936428-08-2.
• Fontaine, Anne; Hurley, Susan (2011). "Golden Matrix Families". The College Mathematics Journal. 42 (2): 140–147. doi:10.4169/college.math.j.42.2.140.
• Gould, Henry W. (1981). "A History of the Fibonacci Q-Matrix and a Higher-Dimensional Problem" (PDF). The Fibonacci Quarterly. 19 (3): 250–257. doi:10.1080/00150517.1981.12430088.
• Hardy, G. H.; Wright, E. M. (1954). "Quadratic Fields (2)". An Introduction to the Theory of Numbers (3rd ed.). Oxford: Clarendon Press. Ch. XV, pp. 218–232.
• Hirzebruch, Friedrich (1976). "The Hilbert modular group for the field ⁠$\mathbf {Q} ({\sqrt {5}}~\!)$⁠, and the cubic diagonal surface of Clebsch and Klein". Russian Mathematical Surveys. 31 (5): 96–110. doi:10.1070/RM1976v031n05ABEH004190. MR 0498397.
• Hoggatt, Verner E. Jr.; Ruggles, Ivan D. (1963). "A Primer for the Fibonacci Numbers – Part IV" (PDF). The Fibonacci Quarterly. 1 (4): 65–71. doi:10.1080/00150517.1963.12431555.
• Hunt, Bruce (1996). "The 27 lines on a cubic surface". The Geometry of some special Arithmetic Quotients. Berlin: Springer. doi:10.1007/BFb0094404. ISBN 3-540-61795-7.
• Huntley, Herbert E. (1970). The Divine Proportion: A Study in Mathematical Beauty. New York: Dover. ISBN 0-486-22254-3.
• Ireland, Kenneth; Rosen, Michael (1990). A Classical Introduction to Modern Number Theory (2nd ed.). New York: Springer. doi:10.1007/978-1-4757-2103-4. ISBN 0-387-97329-X.
• Legendre, Adrien-Marie (1830). "De l'équation ⁠$\textstyle x^{5}+y^{5}+z^{5}=0$⁠". Théorie des Nombres. Vol. II (3rd ed.). Paris: Firmin Didot Fréres. § 6.IV pp. 361–368.
• Lehrer, Gustav I.; Taylor, Donald E. (2009). Unitary Reflection Groups. Cambridge University Press. ISBN 978-0-521-74989-3.
• LeVeque, William J. (1956). "Algebraic Numbers". Topics in Number Theory. Vol. 2. Reading, MA: Addison-Wesley. Ch. 2, pp. 34–81. LCCN 56-10138.
• Liba, Opher; Ilany, Bat-Sheva (2023). From the Golden Rectangle to the Fibonacci Sequences. Contributions by Nativ, Isaac. Cham: Springer. doi:10.1007/978-3-030-97600-2 (inactive 19 Jul 2025). ISBN 978-3-030-97599-9.{{cite book}}: CS1 maint: DOI inactive as of July 2025 (link) Originally published as ממלבן הזהב לסדרות פיבונאצי. Tel Aviv: Mofet Institute. 2019.
• Lind, Douglas A. (1968). "The quadratic field ⁠$\mathbf {Q} ({\sqrt {5}}~\!)$⁠ and a certain Diophantine equation" (PDF). The Fibonacci Quarterly. 6 (3): 86–93. doi:10.1080/00150517.1968.12431231.
• Méndez-Delgadillo, Hugo; Lam-Estrada, Pablo; Maldonado-Ramírez, Myriam R. (2015). "Approach to Square Roots Applying Square Matrices". Palestine Journal of Mathematics. 4 (2).
• Pleasants, Peter A. B. (2002). "Lines and Planes in 2- and 3-Dimensional Quasicrystals". In Kramer, Peter; Papadopolos, Zorka (eds.). Coverings of Discrete Quasiperiodic Sets. Berlin: Springer. pp. 185–225. doi:10.1007/3-540-45805-0_6. ISBN 3-540-43241-8.
• Polo-Blanco, Irene; Top, Jaap (2009). "A remark on parameterizing nonsingular cubic surfaces". Computer Aided Geometric Design. 26 (8): 842–849. doi:10.1016/j.cagd.2009.06.001.
• Ribenboim, Paolo (1999). "The Quintic Equation". Fermat's Last Theorem for Amateurs. New York: Springer. § I.6, pp. 49–57. doi:10.1007/978-0-387-21692-8_2. ISBN 0-387-98508-5.
• Rokhsar, Daniel S.; Mermin, N. David; Wright, David C. (1987). "Rudimentary quasicrystallography: The icosahedral and decagonal reciprocal lattices". Physical Review B. 35 (11): 5487–5495. doi:10.1103/physrevb.35.5487.
• Rotman, Joseph J. (2017) [2002]. "Algebraic Integers". Advanced Modern Algebra. Vol. 2. American Mathematical Society. § C-5.3.2, pp. 455–467. ISBN 978-1-4704-2311-7.
• Sloane, Neil J. A. (ed.). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  • A000032, "Lucas numbers beginning at ⁠$2$⁠: ⁠$L(n)=L(n-1)+L(n-2)$⁠, ⁠$L(0)=2$⁠, ⁠$L(1)=1$⁠"
  • A000045, "Fibonacci numbers: ⁠$F(n)=F(n-1)+F(n-2)$⁠ with ⁠$F(0)=0$⁠ and ⁠$F(1)=1$⁠"
  • A001622, "Decimal expansion of golden ratio ⁠$\varphi$⁠ (or ⁠$\tau$⁠) ⁠$=(1+{\sqrt {5}}~\!)/2$⁠"
  • A002163, "Decimal expansion of square root of ⁠$5$⁠"
  • A003172, "⁠$\mathbf {Q} ({\sqrt {n}})$⁠ is a unique factorization domain (or simple quadratic field)"
  • A031363, "Positive numbers of the form ⁠$\textstyle x^{2}+xy-y^{2}$⁠; or, of the form ⁠$\textstyle 5x^{2}-y^{2}$⁠"
  • A039834, "⁠$a(n+2)=-a(n+1)+a(n)$⁠ (signed Fibonacci numbers) with ⁠$a(-2)=a(-1)=1$⁠; or Fibonacci numbers (A000045) extended to negative indices"
  • A048981, "Squarefree values of ⁠$n$⁠ for which the quadratic field ⁠$\mathbf {Q} [{\sqrt {n}}~\!]$⁠ is norm-Euclidean"
• Sporn, Howard (2021). "A group structure on the golden triples". The Mathematical Gazette. 105 (562): 87–97. doi:10.1017/mag.2021.11.
• Steeb, Willi-Hans; Hardy, Yorick; Tanski, Igor (2012). Problems And Solutions For Groups, Lie Groups, Lie Algebras With Applications. Singapore: World Scientific. ISBN 978-981-310-411-2.
• Trifković, Mak (2013). Algebraic Theory of Quadratic Numbers. New York: Springer. doi:10.1007/978-1-4614-7717-4. ISBN 978-1-4614-7716-7.
• Vajda, Steven (1989). Fibonacci & Lucas Numbers, and the Golden Section: Theory and Applications. Chichester: Ellis Horwood. ISBN 0-7458-0715-1.
• Wasteels, Joseph (1902). "Quelques propriétés des nombres de Fibonacci". Mathesis. 3. 11: 60–62.