Golden field

In mathematics, Template:Q(√5), sometimes called the golden field, is a number system consisting of the set of all numbers Template:Tmath, where Template:Tmath and Template:Tmath are both rational numbers and Template:Tmath is the square root of 5, along with the basic arithmetical operations (addition, subtraction, multiplication, and division). This arithmetic shares certain structural properties with the arithmetic of Template:Tmath, the field of rational numbers, making Template:Q(√5) also a type of field. More specifically, it is a real quadratic field, the extension field of Template:Tmath generated by combining rational numbers and Template:Tmath using arithmetical operations. The name comes from the golden ratio Template:Tmath, a positive number satisfying the equation Template:Tmath, which is the fundamental unit of Template:Q(√5).

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 Template:Tmath where Template:Tmath and Template:Tmath are uniquely determinedTemplate:Sfn rational numbers, or in the form Template:Tmath where Template:Tmath, Template:Tmath, and Template:Tmath are integers, which can be uniquely reduced to lowest terms, and where Template:Tmath is the square root of 5. It is sometimes more convenient instead to use the form Template:Tmath where Template:Tmath and Template:Tmath are rational or the form Template:Tmath where Template:Tmath, Template:Tmath, and Template:Tmath are integers, and where $\phi =\frac{1}{2}(1+\sqrt{5})=$$1.618\dots$ is the golden ratio.Template:Sfn

Converting between these alternative forms is straight-forward: Template:Tmath or, in the other direction, Template:Tmath.Template:Sfn

To add or subtract two numbers, simply add or subtract the components separately:Template:Sfn $$\begin{array}{cc}(a_1+b_1\sqrt{5})+(a_2+b_2\sqrt{5})& =(a_1+a_2)+(b_1+b_2)\sqrt{5},\\ (a_1+b_1\phi )+(a_2+b_2\phi )& =(a_1+a_2)+(b_1+b_2)\phi .\end{array}$$

To multiply two numbers, distribute:Template:Sfn $$\begin{array}{cc}(a_1+b_1\sqrt{5})(a_2+b_2\sqrt{5})& =(a_1{a}_2+5b_1{b}_2)+(a_1{b}_2+b_1{a}_2)\sqrt{5},\\ (a_1+b_1\phi )(a_2+b_2\phi )& =(a_1{a}_2+b_1{b}_2)+(a_1{b}_2+b_1{a}_2+b_1{b}_2)\phi .\end{array}$$

To find the reciprocal of a number $\alpha$, rationalize the denominator: $1/\alpha =$$\bar{\alpha }/\alpha \bar{\alpha }=$Template:Nobr where Template:Tmath is the algebraic conjugate and Template:Tmath is the field norm, as defined below.Template:Sfn Explicitly: $$\begin{array}{cc}\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},\\ \frac{1}{a+b\phi }& =\frac{1}{a+b\phi }\cdot \frac{a+b-b\phi }{a+b-b\phi }=\frac{a+b}{a^2+ab-b^2}-\frac{b}{a^2+ab-b^2}\phi .\end{array}$$

To divide two numbers, multiply the first by second's reciprocal:Template:Sfn $$\begin{array}{cc}\frac{a_1+b_1\sqrt{5}}{a_2+b_2\sqrt{5}}& =\frac{a_1{a}_2-5b_1{b}_2}{a_2^2-5b_2^2}+\frac{-a_1{b}_2+b_1{a}_2}{a_2^2-5b_2^2}\sqrt{5},\\ \frac{a_1+b_1\phi }{a_2+b_2\phi }& =\frac{a_1{a}_2+a_1{b}_2-b_1{b}_2}{a_2^2+a_2{b}_2-b_2^2}+\frac{-a_1{b}_2+b_1{a}_2}{a_2^2+a_2{b}_2-b_2^2}\phi .\end{array}$$

As in any field, addition and multiplication of numbers in Template:Q(√5) is associative and commutative; Template:Tmath is the additive identity and Template:Tmath is the multiplicative identity; every number Template:Tmath has an additive inverse Template:Tmath and a multiplicative inverse Template:Tmath; and multiplication distributes over addition. Arithmetic between numbers in Template:Q(√5) is consistent with their arithmetic as real numbers; that is, Template:Q(√5) is a subfield of Template:Tmath.

Conjugation and norm

The numbers Template:Tmath and Template:Tmath each solve the equation Template:Tmath. Each number Template:Tmath in Template:Q(√5) has an algebraic conjugate Template:Tmath found by swapping these two square roots of 5, i.e., by changing the sign of Template:Tmath. The conjugate of Template:Tmath is $\bar{\phi }=\frac{1}{2}(1-\sqrt{5})=$$-{\phi }^{-1}=$Template:Nobr. A rational number is its own conjugate, Template:Tmath. In general, the conjugate is:Template:Sfn $$\begin{array}{cc}\overline{a+b\sqrt{5}}& =a-b\sqrt{5},\\ \overline{a+b\phi }& =a+b\bar{\phi }=(a+b)-b\phi .\end{array}$$ Conjugation in Template:Q(√5) is an involution, Template:Tmath, and it preserves the structure of arithmetic: Template:Tmath; Template:Tmath; and Template:Tmath.Template:Sfn Conjugation is the only ring homomorphism (function preserving the structure of addition and multiplication) from Template:Q(√5) 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).Template:Sfn The trace of Template:Tmath in Template:Q(√5) is Template:Tmath: $$\begin{array}{cc}\mathrm{t}\mathrm{r}(a+b\sqrt{5})& =(a+b\sqrt{5})+(a-b\sqrt{5})=2a,\\ \mathrm{t}\mathrm{r}(a+b\phi )& =(a+b\phi )+(a+b-b\phi )=2a+b.\end{array}$$ This is always an (ordinary) rational number.Template:Sfn

The field norm is a measure of a number's magnitude, the product of the number and its conjugates.Template:Sfn The norm of Template:Tmath in Template:Q(√5) is Template:Tmath:Template:Sfn $$\begin{array}{cc}\mathrm{N}(a+b\sqrt{5})& =a^2-5b^2,\\ \mathrm{N}(a+b\phi )& =a^2+ab-b^2.\end{array}$$ This is also always a rational number.Template:Sfn

The norm preserves the structure of multiplication, as expected for a concept of magnitude. The norm of a product is the product of norms, Template:Tmath; and the norm of a quotient is the quotient of the norms, Template:Tmath. A number and its conjugate have the same norm, Template:Tmath.Template:Sfn

A number Template:Tmath in Template:Q(√5) and its conjugate Template:Tmath are the solutions of the quadratic equationTemplate:Sfn $$(x-\alpha )(x-\bar{\alpha })=x^2-\mathrm{t}\mathrm{r}(\alpha )x+\mathrm{N}(\alpha )=0.$$

In Galois theory, the golden field can be considered more abstractly as the set of all numbers Template:Tmath, where Template:Tmath and Template:Tmath are both rational, and all that is known of Template:Tmath is that it satisfies the equation Template:Tmath. There are two ways to embed this set in the real numbers: by mapping Template:Tmath to the positive square root Template:Tmath or alternatively by mapping Template:Tmath to the negative square root Template:Tmath. 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.Template:Sfn

Golden integers

One convenient way to plot Template:Math is as a lattice in the real coordinate plane. The so-called canonical embedding represents the number Template:Math by the point with coordinates Template:Math. Numbers with the same norm lie on hyperbolas (orange and green lines).

The logarithmic embedding into the plane represents the (non-zero) number Template:Math by the point with coordinates Template:Math – See Log–log plot. Multiplication in the golden field corresponds to vector addition in the embedding. Numbers whose norm has the same absolute value lie on diagonal lines.

The ring of integers of the golden field, Template:Tmath, sometimes called the golden integers, is the set of all algebraic integers in the field, meaning those elements whose minimal polynomial over Template:Tmath has integer coefficients. These are the set of numbers in Template:Q(√5) whose norm and trace are integers. The numbers Template:Tmath and Template:Tmath form an integral basis for the ring, meaning every number in the ring can be written in the form $a+b\phi$ where $a$ and $b$ are ordinary integers.Template:Sfnm Alternately, elements of Template:Tmath can be written in the form Template:Tmath, where Template:Tmath and Template:Tmath have the same parity.Template:Sfn Like any ring, Template:Tmath is closed under addition and multiplication. Template:Q(√5) is the smallest field containing Template:Tmath, its field of fractions.

Norms

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

Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath,Template:Nobr

The golden integer Template:Tmath is called zero, and is the only element of Template:Tmath with norm Template:Tmath.Template:Sfn

Divisibility

If Template:Tmath and Template:Tmath are golden integers and there is some golden integer Template:Tmath such that Template:Tmath, we say that Template:Tmath divides Template:Tmath, denoted Template:Tmath. In many respects, divisibility works similarly as among the ordinary integers, but with some important differences, as will be described in the following subsections.

Because, like the integers, Template:Tmath is an integral domain, the product of two non-zero elements is always non-zero. Thus Template:Tmath has no nontrivial zero divisors, and Template:Tmath implies that either Template:Tmath or Template:Tmath.

Units

A unit is an algebraic integer which divides Template:Tmath, i.e. whose multiplicative inverse is also an algebraic integer, which happens when its norm is Template:Tmath. Among the ordinary integers, the units are the pair of numbers Template:Tmath, but among the golden integers there are infinitely many units: all numbers of the form Template:Tmath whose integer coefficients Template:Tmath and Template:Tmath solve the Diophantine equation Template:Tmath. If a unit is instead written in the form Template:Tmath, its coefficients solve a related Diophantine equation, the generalized Pell's equation Template:Tmath. The fundamental unit – the smallest unit greater than Template:Tmath – is the golden ratio Template:Tmath and the other units consist of its positive and negative powers, Template:Tmath, for any integer Template:Tmath.Template:Sfn Some [[Golden ratio#Golden ratio conjugate and powers|powers of Template:Tmath]] are:

Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath

In general Template:Tmath, where Template:Tmath is the Template:Tmathth Fibonacci number. The units form the group Template:Tmath under multiplication, which can be decomposed as the direct product of a cyclic group of order 2 generated by Template:Tmath and an infinite cyclic group generated by Template:Tmath.

Associates

Two golden integers Template:Tmath and Template:Tmath are associates if each divides the other, Template:Tmath and Template:Tmath. Equivalently, their quotient in Template:Tmath is a unit, Template:Tmath for some integer Template:Tmath. Associateness is an equivalence relation. If Template:Tmath divides some golden integer Template:Tmath, then so does its associate Template:Tmath: if Template:Tmath then Template:Tmath.

Associates have the same norm, up to sign: Template:Nobr However, not all elements whose norm has the same absolute value are associates; in particular, any golden-integer prime and its conjugate have the same norm, but are associates if and only if they are associated either with Template:Tmath or with an ordinary prime.Template:Sfn

More generally, two numbers in Template:Q(√5) are associates if their quotient is a unit. The set of associates of any number in Template:Q(√5) is the orbit of any of them under the multiplicative action of the group of units.

Primes and prime factorization

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

A prime element of a ring, analogous to a prime number among the integers, is an element Template:Tmath such that whenever Template:Tmath, then either Template:Tmath or Template:Tmath. In Template:Tmath the primes are of three types: Template:Tmath, integer primes of the form Template:Tmath where Template:Tmath is an integer, and the factors of integer primes of the form Template:Tmath (a pair of conjugates).Template:Sfn For example, Template:Tmath, Template:Tmath, and Template:Tmath are primes, but Template:Tmath is composite. Any of these is an associate of additional primes; for example Template:Tmath is also prime, an associate of Template:Tmath.Template:R

The ring Template:Tmath 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.Template:Sfn This makes Template:Q(√5) one of the 21 quadratic fields that are norm-Euclidean. A form of the fundamental theorem of arithmetic applies in Template:Tmath: every golden integer 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 one of its associates (which changes the unit factor accordingly).

Ideals

An ideal of Template:Tmath is any subset which "absorbs multiplication", containing every golden-integer multiple of each of its elements. If Template:Tmath is any golden integer, the set of all golden-integer multiples of Template:Tmath, denoted Template:Tmath or Template:Tmath, is the ideal generated by Template:Tmath. Every associated element generates the same ideal, but a non-associated element generates a different ideal: that is, Template:Tmath precisely when Template:Tmath. Because Template:Tmath is a principal ideal domain, each ideal of Template:Tmath can be generated by a single element. The zero ideal Template:Tmath is the single-element set Template:Tmath. The ideal Template:Tmath is all of Template:Tmath.

Various operations can be defined among ideals. If Template:Tmath and Template:Tmath are ideals of Template:Tmath, then Template:Tmath is the set of all sums of one element in Template:Tmath plus one element in Template:Tmath, and Template:Tmath is the set of all sums of any number of terms, each of which is the product of one element in Template:Tmath times one element in Template:Tmath. $$\begin{array}{cc}𝔞+𝔟& =\{\alpha +\beta \mid \alpha \in 𝔞,\beta \in 𝔟\},\\ 𝔞𝔟& =\{{\sum }_{i=1}^n{\alpha }_i{\beta }_i|n\in \mathbb{N},a_i\in 𝔞,b_i\in 𝔟\}.\end{array}$$ The sum or product of two ideals is itself an ideal. Multiplication of ideals is distributive over addition.

More generally, a fractional ideal of Template:Q(√5) is a subset of Template:Q(√5) with the property that multiplication of each element by some golden integer, the "denominator", results in an ideal of Template:Tmath. If Template:Tmath is any number in Template:Q(√5), the set of golden-integer multiples of Template:Tmath, also denoted Template:Tmath or Template:Tmath, is the fractional ideal generated by Template:Tmath. As with integral ideals of Template:Tmath, numbers in Template:Q(√5) generate the same fractional ideal if and only if they are associated, and every fractional ideal can be generated by a single element. Multiplication of fractional ideals is consistent with multiplication of their generators. Let Template:Tmath be the multiplicative group of the nonzero elements of Template:Q(√5), and Template:Tmath be the group of the units. The function that maps each element of Template:Tmath to the generated fractional ideal, Template:Tmath, induces a group isomorphism between Template:Tmath and the group of fractional ideals.

Table of integers

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 Template:Tmath in the class for which $\alpha +|\bar{\alpha }|$ is a minimum).Template:R

Rep.	Norm	Trace	Factorization	Positive associates
Template:Tmath	Template:Tmath	Template:Tmath	zero
Template:Tmath	Template:Tmath	Template:Tmath	unit	Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	prime	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	prime	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	prime	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	prime	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	prime	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	prime	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	prime	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	prime	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	prime	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	prime	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	prime	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	prime	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	prime	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	prime	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	prime	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	prime	Template:Tmath, Template:Tmath, Template:Tmath; Template:Tmath, Template:Tmath, Template:Tmath
Template:Tmath Show/hide more rows

Matrix representation

Template:Q(√5) is a two-dimensional vector space over Template:Tmath, and multiplication by any element of Template:Q(√5) is a linear transformation of that vector space. Given an ordered basis of Template:Q(√5), each number in Template:Q(√5) can be associated to the corresponding transformation matrix in that basis. This defines a field isomorphism (a structure-preserving bijective map) from Template:Q(√5) to the space of Template:Tmath square matrices with rational entries spanned by the identity matrix Template:Tmath, the image of the number Template:Tmath, and a matrix Template:Tmath, the image of Template:Tmath.Template:R Thus arithmetic of numbers in Template:Q(√5) can be alternately represented by the arithmetic of such matrices. In this context, the number Template:Tmath is represented by the matrix Template:Tmath.Template:Sfn A convenient choice of basis for Template:Q(√5) is Template:Tmath, in terms of which Template:Tmath is a symmetric matrix: $$\begin{array}{cc}& 𝐈=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],𝜱=\left[\begin{array}{cc}0& 1\\ 1& 1\end{array}\right],𝐀=a𝐈+b𝜱=\left[\begin{array}{cc}a& b\\ b& a+b\end{array}\right].\end{array}$$

The adjugate matrix Template:Tmath represents the algebraic conjugate Template:Tmath, the matrix Template:Tmath (satisfying Template:Tmath) represents Template:Tmath, and the adjugate of an arbitrary element Template:Tmath, which we will denote Template:Tmath, represents the number Template:Tmath: $$\begin{array}{cc}& 𝐑=\left[\begin{array}{cc}-1& 2\\ 2& 1\end{array}\right],\overline{𝜱}=\left[\begin{array}{cc}1& -1\\ -1& 0\end{array}\right],\overline{𝐀}=a𝐈+b\overline{𝜱}=\left[\begin{array}{cc}a+b& -b\\ -b& a\end{array}\right].\end{array}$$

Every matrix Template:Tmath, except for the zero matrix, is invertible, and its inverse Template:Tmath represents the multiplicative inverse Template:Tmath in Template:Q(√5).Template:Sfn

If Template:Tmath is an element of Template:Q(√5), with conjugate Template:Tmath, then the matrix Template:Tmath has the numbers Template:Tmath and Template:Tmath as its eigenvalues. Its trace is $\mathrm{t}\mathrm{r}(𝐀)=$$2a+b=$Template:Nobr Its determinant is $\mathrm{d}\mathrm{e}\mathrm{t}(𝐀)=$$a^2+ab-b^2=$Template:Nobr The characteristic polynomial of Template:Tmath is Template:Tmath, which is the minimal polynomial of Template:Tmath and Template:Tmath whenever Template:Tmath is not zero. These properties are shared by the adjugate matrix Template:Tmath. Their product is Template:Tmath.Template:R

These matrices have especially been studied in the context of the Fibonacci numbers Template:Tmath and Lucas numbers Template:Tmath, which appear as the entries of Template:Tmath and Template:Tmath, respectively: $$\begin{array}{cccc}{𝜱}^n& =F_{n-1}𝐈+F_n𝜱=\left[\begin{array}{cc}{F}_{n-1}& F_n\\ F_n& F_{n+1}\end{array}\right],& \mathrm{t}\mathrm{r}({𝜱}^n)& =L_n,\\ {𝜱}^n𝐑& =L_{n-1}𝐈+L_n𝜱=\left[\begin{array}{cc}{L}_{n-1}& L_n\\ L_n& L_{n+1}\end{array}\right],& \mathrm{t}\mathrm{r}({𝜱}^n𝐑)& =5F_n.\end{array}$$ Powers of Template:Tmath are sometimes called Fibonacci matrices.

Every matrix of the form Template:Tmath has eigenvectors which point along the directions Template:Tmath and Template:Tmath. When numbers in Template:Q(√5) 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 Template:Q(√5) directly on either axis.) The matrices Template:Tmath for integer Template:Tmath, representing units, and more generally any matrices with Template:Tmath and determinant Template:Tmath, are squeeze mappings, which stretch the plane along one axis and squish it along the other, fixing hyperbolas of constant norm. The matrices Template:Tmath and more generally matrices with Template:Tmath and determinant Template:Tmath, are the composition of a squeeze mapping and a vertical reflection. The negative identity matrix Template:Tmath is a point reflection across the origin. In general any other matrix Template:Tmath 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, Template:Tmath.Template:Sfn It has class number 1, which means that the ring of its algebraic integers is a principal ideal domain and 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 $\phi$.Template:Sfn

Fibonacci numbers

The Lucas and Fibonacci numbers are components of Template:Math when written in terms of Template:Math and Template:Math.

Template:Q(√5) is a natural choice of number system for studying the Fibonacci numbers Template:Tmath and the Lucas numbers Template:Tmath. These number sequences are usually defined by recurrence relations similar to the one satisfied by the powers of Template:Tmath and Template:Tmath: $$\begin{array}{cccccc}{F}_0& =0,& F_1& =1,& F_{n+1}& =F_n+F_{n-1},\\ L_0& =2,& L_1& =1,& L_{n+1}& =L_n+L_{n-1},\\ {\phi }^0& =1,& {\phi }^1& =\phi ,& {\phi }^{n+1}& ={\phi }^n+{\phi }^{n-1},\\ {\bar{\phi }}^0& =1,& {\bar{\phi }}^1& =\bar{\phi },& {\bar{\phi }}^{n+1}& ={\bar{\phi }}^n+{\bar{\phi }}^{n-1}.\end{array}$$

The sequences Template:Tmath and Template:Tmath respectively begin:

Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath
Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath	Template:Tmath

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

The Fibonacci and Lucas numbers can alternately be expressed as the components Template:Tmath and Template:Tmath when a power of the golden ratio or its conjugate is written in the form Template:Tmath: $$\begin{array}{cc}{\phi }^n& =\frac{1}{2}{L}_n+\frac{1}{2}{F}_n\sqrt{5},\\ {\bar{\phi }}^n& =\frac{1}{2}{L}_n-\frac{1}{2}{F}_n\sqrt{5}.\end{array}$$

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

The expression of the Fibonacci numbers in terms of Template:Tmath is called Binet's formula:

$$\begin{array}{cc}{F}_n& =\frac{{\phi }^n-{\bar{\phi }}^n}{\phi -\bar{\phi }}=\frac{{\phi }^n-{\bar{\phi }}^n}{\sqrt{5}}=\frac{\mathrm{t}\mathrm{r}\left({\phi }^n\sqrt{5}\right)}{5},\\ L_n& =\frac{{\phi }^n+{\bar{\phi }}^n}{\phi +\bar{\phi }}={\phi }^n+{\bar{\phi }}^n=\mathrm{t}\mathrm{r}\left({\phi }^n\right).\end{array}$$

The powers of Template:Tmath or Template:Tmath, when written in the form Template:Tmath, can be expressed in terms of just Fibonacci numbers,Template:Sfn $$\begin{array}{cc}{\phi }^n& =F_{n-1}+F_n\phi ,\\ {\bar{\phi }}^n& =F_{n-1}+F_n\bar{\phi }=F_{n+1}-F_n\phi .\end{array}$$ Powers of Template:Tmath or Template:Tmath times Template:Tmath can be expressed in terms of just Lucas numbers, $$\begin{array}{cc}{\phi }^n\sqrt{5}& =L_{n-1}+L_n\phi ,\\ {\bar{\phi }}^n\sqrt{5}& =-L_{n-1}-L_n\bar{\phi }=-L_{n+1}+L_n\phi .\end{array}$$ Statements about golden integers can be recast as statements about the Fibonacci or Lucas numbers; for example, that every power of Template:Tmath is a unit of Template:Tmath, Template:Tmath, when expanded, becomes Cassini's identity, and likewise Template:Tmath becomes the analogous identity about Lucas numbers, $$\begin{array}{cc}(F_{n-1}+F_n\phi )(F_{n+1}-F_n\phi )& =F_{n-1}{F}_{n+1}-F_n^2=(-1{)}^n,\\ (L_{n-1}+L_n\phi )(-L_{n+1}+L_n\phi )& =L_n^2-L_{n-1}{L}_{n+1}=(-1{)}^{n}5.\end{array}$$

The numbers Template:Tmath and Template:Tmath are the roots of the quadratic polynomial Template:Tmath. This is the minimal polynomial for Template:Tmath for any non-zero integer Template:Tmath. The quadratic polynomial Template:Tmath is the minimal polynomial for Template:Tmath.

In the limit, consecutive Fibonacci or Lucas numbers approach a ratio of Template:Tmath, and the ratio of Lucas to Fibonacci numbers approaches Template:Tmath:Template:R $$\begin{array}{cccc}\underset{n\to \mathrm{\infty }}{lim}\frac{F_{n+1}}{F_n}& =\underset{n\to \mathrm{\infty }}{lim}\frac{L_{n+1}}{L_n}=\phi ,& \underset{n\to \mathrm{\infty }}{lim}\frac{L_n}{F_n}& =\sqrt{5}.\end{array}$$

Theorems about the Fibonacci numbers – for example, divisibility properties such as if Template:Tmath divides Template:Tmath then Template:Tmath divides Template:Tmath – can be conveniently proven using Template:Q(√5).

Relation to fivefold symmetry

The golden ratio Template:Tmath 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 Template:Tmath be the 5th root of unity, a complex number of unit absolute value spaced Template:Tmath of a full turn from Template:Tmath around the unit circle, satisfying Template:Tmath. Then the fifth cyclotomic field Template:Tmath is the field extension of the rational numbers formed by adjoining Template:Tmath (or equivalently, adjoining any of Template:Tmath, Template:Tmath or Template:Tmath). Elements of Template:Tmath are numbers of the form $a_0+a_1\zeta +$$a_2{\zeta }^2+$Template:Nobr, with rational coefficients. Template:Tmath is of degree four over the rational numbers: any four of the five roots are linearly independent over Template:Tmath, but all five sum to zero. However, Template:Tmath is only of degree two over Template:Q(√5), $$\begin{array}{cc}{x}^5-1& =(x-1)(x^4+x^3+x^2+x+1)\\ & =(x-1)(x^2+\bar{\phi }x+1)(x^2+\phi x+1)\\ & =(x-1)(x-\zeta )(x-{\zeta }^4)(x-{\zeta }^2)(x-{\zeta }^3),\end{array}$$ where the conjugate Template:Tmath. The elements of Template:Tmath can alternately be represented as Template:Tmath, where Template:Tmath and Template:Tmath are elements of Template:Q(√5): $$\begin{array}{cc}& a_0+a_1\zeta +a_2{\zeta }^2+a_3{\zeta }^3+a_4{\zeta }^4\\ & =(a_0-a_2+\bar{\phi }{a}_3-\bar{\phi }{a}_4)+(a_1-\bar{\phi }{a}_2+\bar{\phi }{a}_3-a_4)\zeta .\end{array}$$

Conversely, Template:Q(√5) is a subfield of Template:Tmath. For any primitive root of unity Template:Tmath, the maximal real subfield of the cyclotomic field Template:Tmath is the field Template:Tmath; see [[Minimal polynomial of 2cos(2pi/n)|Minimal polynomial of Template:Tmath]]. In our case Template:Tmath, $(\zeta +{\zeta }^{-1}{)}^2+$Template:Nobr so Template:Tmath is the positive root of the quadratic polynomial Template:Tmath, namely Template:Tmath, and the maximal real subfield of Template:Tmath is Template:Q(√5).

Diagonals of a unit-circumradius regular decagon are square roots of golden integers.

The squared absolute value of any element of Template:Tmath is an element of Template:Q(√5): $$\begin{array}{cc}|\alpha +\beta \zeta {|}^2& =(\alpha +\beta \zeta )(\alpha +\beta {\zeta }^{-1})={\alpha }^2+{\beta }^2+\alpha \beta {\phi }^{-1}.\end{array}$$ In particular, the squared length of the side of a regular pentagon of unit circumradius is $|\zeta -1|{}^2=$$2+\bar{\phi }=$${\phi }^{-1}\sqrt{5}$ and the squared length of its diagonal is $|{\zeta }^2-1|{}^2=$$2+\phi =$Template:Nobr these two numbers are conjugates, associates of Template:Tmath. In a regular decagon with unit circumradius, the squared length of a side is $|{\zeta }^2+1|{}^2={\phi }^{-2}$ and the squared length of a diagonal between vertices three apart is Template:Nobr

Golden integers are involved in the trigonometric study of fivefold symmetries. By the quadratic formula, $$\begin{array}{ccc}\zeta & =\frac{1}{2}(-\bar{\phi }+\sqrt{\bar{\phi }{}^2-4})& =-\frac{1}{2}\bar{\phi }+\frac{1}{2}\sqrt{-2-\phi },\\ {\zeta }^2& =\frac{1}{2}(-\phi +\sqrt{\phi {}^2-4})& =-\frac{1}{2}\phi +\frac{1}{2}\sqrt{-2-\bar{\phi }}.\end{array}$$

Angles of Template:Tmath and Template:Tmath thus have golden rational cosines but their sines are the square roots of golden rational numbers. $$\begin{array}{cccc}\cos \frac{2}{5}\pi & =-\frac{1}{2}\bar{\phi },& \sin \frac{2}{5}\pi & =\frac{1}{2}\sqrt{2+\phi },\\ \cos \frac{4}{5}\pi & =-\frac{1}{2}\phi ,& \sin \frac{4}{5}\pi & =\frac{1}{2}\sqrt{2+\bar{\phi }}.\end{array}$$

Three-dimensional space

A regular icosahedron with edge length Template:Tmath can be oriented so that the Cartesian coordinates of its vertices areTemplate:Sfn $$(0,\pm 1,\pm \phi ),(\pm 1,\pm \phi ,0),(\pm \phi ,0,\pm 1).$$

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 Template:Tmath generated by four mirrors which can be conveniently oriented as Template:Tmath, Template:Tmath, Template:Tmath, and Template:Tmath. Then the 120 vertices have golden-integer coordinates: arbitrary permutations of Template:Tmath and Template:Tmath with an even number of minus signs, Template:Tmath with an odd number of minus signs, and Template:Tmath.

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.Template:Sfnm The icosians of unit norm are the vertices of a 600-cell.Template:Sfn

Quasiperiodicity

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

Golden integers are used in studying quasicrystals.Template:Sfn

Other applications

The quintic case of Fermat's Last Theorem, that there are no nontrivial integer solutions to the equation Template:Tmath, was proved using Template:Q(√5) 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.Template:Sfn They can, in fact, be defined over the golden field.Template:Sfnm

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.Template:Sfnm

References

• Template:Cite journal
• Template:Cite web
• Template:Cite journal
• Template:Cite journal
• Template:Cite book
• Template:Cite journal
• Template:Cite journal
• Template:Cite journal
• Template:Cite book
• Template:Cite journal
• Template:Cite journal
• Template:Cite journal
• Template:Cite book
• Template:Cite journal
• Template:Cite journal
• Template:Cite book
• Template:Cite journal
• Template:Cite journal
• Template:Cite book
• Template:Cite journal
• Template:Cite journal
• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Wikicite
• Template:Cite journal
• Template:Cite journal
• Template:Cite book
• Template:Cite journal
• Template:Cite book
• Template:Cite journal
• Template:Cite book
• Template:Cite book
• Template:Wikicite
• Template:Cite journal
• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite journal
• Template:Cite journal