Golden field
In mathematics, the golden field 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 , where and are both rational. The name comes from the golden ratio , which is the fundamental unit of , and which satisfies the equation .
Basic arithmetic
[edit]Elements of the golden field can be written in the form where and are uniquely determined[1] rational numbers, or in the form where , , and are integers, which can be uniquely reduced to lowest terms. It is sometimes more convenient instead to use the form where and are rational and is the golden ratio, or the form where , , and are integers.
Converting between these alternative forms is straight-forward: , or in the other direction .
To add or subtract two numbers, simply add or subtract the components separately:
To multiply two numbers, distribute:
To find the reciprocal of a number, rationalize the denominator:
To divide two numbers, multiply the first by second's reciprocal:
Conjugation and norm
[edit]The numbers and both solve the equation . Each golden rational number has a Galois conjugate found by swapping these two square roots of 5, i.e., by changing the sign of . The conjugate of is . Because the conjugate of is , the conjugate of is .
Multiplying a number in 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 is:
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, ; and the norm of a number's reciprocal is the reciprocal of the norm, .
Properties
[edit]The ring of integers of , sometimes called the golden integers, is the set of golden rational numbers with integral norm, or equivalently the set of numbers of the form where and are both ordinary integers.[2] A unit is an algebraic integer whose multiplicative inverse is also an algebraic integer, which happens when its norm is . The units of are given by powers of the golden ratio:[1]
The golden field is the real quadratic field with the smallest discriminant.[3] It has class number 1 and is a unique factorization domain.[4]
The golden field is Euclidean: every non-negative element of is the square of some element in .[3]
Applications
[edit]The vertices of the four-dimensional polytope known as the 600-cell can be chosen so that their coordinates lie within the golden integers.[5]
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.[6] The icosians of unit norm are the vertices of a 600-cell.[5]
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.[7] They can, in fact, be defined over the golden field.[8]
Golden integers are used in studying quasicrystals.[9] An abelian extension of the golden field is used in a construction of a SIC-POVM in four-dimensional complex vector space.[10]
Notes
[edit]- ^ a b Lind 1968.
- ^ Hirzebruch 1976; Sporn 2021.
- ^ a b Dembélé 2005.
- ^ Sloane "A003172".
- ^ a b Denney et al. 2020.
- ^ Conway & Sloane 1999, pp. 207–208; Pleasants 2002, pp. 213–214.
- ^ Baez 2016.
- ^ Hunt 1996; Polo-Blanco & Top 2009.
- ^ Sporn 2021.
- ^ Appleby et al. 2022; Bengtsson 2017.
References
[edit]- Appleby, Marcus; Bengtsson, Ingemar; Grassl, Markus; Harrison, Michael; McConnell, Gary (2022). "SIC-POVMs from Stark units: Prime dimensions n2 + 3". Journal of Mathematical Physics. 63: 112205. arXiv:2112.05552. doi:10.1063/5.0083520.
- Baez, John (2016-03-01). "Clebsch Surface". Visual Insight. 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.
- Conway, J. H.; Sloane, N. J. A. (1999). "Algebraic Constructions for Lattices". Sphere Packings, Lattices and Groups. pp. 206–244. doi:10.1007/978-1-4757-6568-7_8.
- Dembélé, Lassina (2005). "Explicit computations of Hilbert modular forms on " (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.
- Hirzebruch, Friedrich (1976). "The Hilbert modular group for the field Q(√5), and the cubic diagonal surface of Clebsch and Klein". Russian Mathematical Surveys. 31 (5): 96–110. doi:10.1070/RM1976v031n05ABEH004190. MR 0498397.
- Hunt, B. (1996). "The 27 lines on a cubic surface". The Geometry of some special Arithmetic Quotients. Lecture Notes in Mathematics. Vol. 1637. Springer. doi:10.1007/BFb0094404. ISBN 978-3-540-61795-2.
- Lind, D. A. (1968). "The quadratic field Q(√5) and a certain Diophantine equation" (PDF). The Fibonacci Quarterly. 6 (3): 86–93. doi:10.1080/00150517.1968.12431231.
- Pleasants, Peter A. B. (2002). "Lines and Planes in 2- and 3-Dimensional Quasicrystals". Coverings of Discrete Quasiperiodic Sets. Springer Tracts in Modern Physics. Vol. 180. Springer. pp. 185–225. doi:10.1007/3-540-45805-0_6. ISBN 978-3-540-43241-8.
- Polo-Blanco, I.; Top, J. (2009). "A remark on parameterizing nonsingular cubic surfaces". Computer Aided Geometric Design. 26 (8): 842–849. doi:10.1016/j.cagd.2009.06.001.
- Sloane, N. J. A. (ed.). "Sequence A003172 (Q(sqrt n) is a unique factorization domain (or simple quadratic field))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- Sporn, Howard (2021). "A group structure on the golden triples". The Mathematical Gazette. 105 (562): 87–97. doi:10.1017/mag.2021.11.