Abstract
Arithmetical algorithms considered in Chap. 5 are based on the arithmetical operations with matrices of the number systems. If the entries of these matrices are not integers or rationals, we need arithmetical algorithms which work with them. Such algorithms exist for algebraic numbers. Algebraic numbers can be represented by vectors or matrices of rational numbers. Arithmetical operations with them are based on matrix calculus.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
van der Waerden, B.L.: Algebra, vol. I. Springer, Berlin (2003)
Ireland, K., Rosen, M.: A classical introduction to modern number theory. Graduate Texts in Mathematics. Springer, Berlin (1990)
Micciancio, D., Goldwasser, S.: Complexity of Lattice Problems: A Cryptographic Perspective. Kluwer Academic Publishers (2002)
Rényi, A.: Representations for real numbers and their ergodic properties. Acta Mathematica Academiae Scientiarum Hungaricae 8, 477–493 (1957)
Parry, W.: On the \(\beta \)-expansions of real numbers. Acta Mathematica Academiae Scientiarum Hungaricae 11, 401–416 (1960)
Schmidt, K.: On periodic expansions of Pisot numbers and Salem numbers. Bull. Lond. Math. Soc. 12(4), 269–278 (1980)
Avizienis, A.: Signed-digit number representations for fast parallel arithmetic. IRE Trans. Electron. Comput. EC-10, 389–400 (1961)
Chow, C.Y., Robertson, J.E.: Logical design of a redundant binary adder. In: IEEE 4th Symposium on Computer Arithmetic, pp. 109–115. IEEE Computer Society (1978)
Frougny, Ch., Pelantová, E., Svobodová, M.: Parallel addition in non-standard numeration systems. Theor. Comput. Sci. 412, 5714–5727 (2011)
Frougny, Ch., Heller, P., Pelantová, E., Svobodová, M.: \(k\)-block parallel addition versus 1-block parallel addition in non-standard numeration systems. Theor. Comput. Sci. 543, 52–67 (2014)
Frougny, Ch., Pelantová, E., Svobodová, M.: Minimal digit sets for parallel addition in non-standard numeration systems. J. Integer Sequences 16(2), 1–36 (2013). Article 13.2.17
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Kůrka, P. (2016). Algebraic Number Fields. In: Dynamics of Number Systems. Studies in Systems, Decision and Control, vol 59. Springer, Cham. https://doi.org/10.1007/978-3-319-33367-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-33367-0_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-33366-3
Online ISBN: 978-3-319-33367-0
eBook Packages: EngineeringEngineering (R0)