Golden field

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

Basic arithmetic

Elements of the golden field 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. It is sometimes more convenient instead to use the form ⁠$a+b\varphi$⁠ where ⁠$a$⁠ and ⁠$b$⁠ are rational and ⁠$\varphi$⁠ is the golden ratio, or the form ⁠$(a+b\varphi )/c$⁠ where ⁠$a$⁠, ⁠$b$⁠, and ⁠$c$⁠ are integers.

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

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

Conjugation and norm

The numbers ⁠${\sqrt {5}}$⁠ and ⁠$-{\sqrt {5}}$⁠ both solve the equation $x^{2}=5$. Each golden rational number ⁠$a+b{\sqrt {5}}$⁠ has a Galois conjugate found by swapping these two square roots of 5, i.e., by changing the sign of ⁠$b$⁠. The conjugate of ⁠$a+b{\sqrt {5}}$⁠ is ⁠$a-b{\sqrt {5}}$⁠. Because the conjugate of ⁠$\varphi$⁠ is $\textstyle {\tfrac {1}{2}}{\bigl (}1-{\sqrt {5}}~\!{\bigr )}={}$$-\varphi ^{-1}={}$$1-\varphi$, the conjugate of ⁠$a+b\varphi$⁠ is ⁠$a-b\varphi ^{-1}=a+b-b\varphi$⁠.

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

The norm has some properties expected for magnitudes. For instance, a number and its conjugate have the same norm; the norm of a product is the product of norms, ⁠$N(xy)=N(x)N(y)$⁠; and the norm of a number's reciprocal is the reciprocal of the norm, ⁠$N(1/x)=1/N(x)$⁠.

Properties

The ring of integers of ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠, sometimes called the golden integers, is the set of golden rational numbers with integral norm, or equivalently the set of numbers of the form $a+b\varphi$ where $a$ and $b$ are both ordinary integers. A unit is an algebraic integer whose multiplicative inverse is also an algebraic integer, which happens when its norm is ⁠$\pm 1$⁠. The units of ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠ are given by powers of the golden ratio: $$\pm \varphi ^{n},\pm \varphi ^{-n},n=0,1,2,3,\ldots$$

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

The golden field is Euclidean: every non-negative element of ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠ is the square of some element in ⁠$\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$⁠.

Applications

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

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.

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.

Golden integers are used in studying quasicrystals. An abelian extension of the golden field is used in a construction of a SIC-POVM in four-dimensional complex vector space.

References

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