Abstract
A computable ring is a ring equipped with a mechanical procedure to add and multiply elements. In most natural computable integral domains, there is a computational procedure to determine if a given element is prime/irreducible. However, there do exist computable UFDs (in fact, polynomial rings over computable fields) where the set of prime/irreducible elements is not computable. Outside of the class of UFDs, the notions of irreducible and prime may not coincide. We demonstrate how different these concepts can be by constructing computable integral domains where the set of irreducible elements is computable while the set of prime elements is not, and vice versa. Along the way, we will generalize Kronecker’s method for computing irreducibles and factorizations in \(\mathbb {Z}[x]\).
The authors thank Grinnell College for its generous support through the MAP program for research with undergraduates. They also thank the referee for providing several corrections and helpful suggetions.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Anderson, D.D., Anderson, D.F., Zafrullah, M.: Factorization in integral domains. J. Pure Appl. Algebra 69(1), 1–19 (1990)
Anderson, D.D., Anderson, D.F., Zafrullah, M.: Factorization in integral domains. II. J. Algebra 152(1), 78–93 (1992)
Anderson, D.D., Mullins, B.: Finite factorization domains. Proc. Am. Math. Soc. 124(2), 389–396 (1996)
Dzhafarov, D., Mileti, J.: The complexity of primes in computable UFDs. Notre Dame J. Form. Log. To appear
Fröhlich, A., Shepherdson, J.C.: Effective procedures in field theory. Philos. Trans. R. Soc. Lond. Ser. A. 248, 407–432 (1956)
Metakides, G., Nerode, A.: Effective content of field theory. Ann. Math. Log. 17(3), 289–320 (1979)
Miller, R.: Computable fields and Galois theory. Not. Am. Math. Soc. 55(7), 798–807 (2008)
Rabin, M.: Computable algebra, general theory and theory of computable fields. Trans. Am. Math. Soc. 95, 341–360 (1960)
Stoltenberg-Hansen, V., Tucker, J.V.: Computable Rings and Fields. Handbook of Computability Theory. Elsevier, Amsterdam (1999)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Evron, L., Mileti, J.R., Ratliff-Crain, E. (2017). Irreducibles and Primes in Computable Integral Domains. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds) Computability and Complexity. Lecture Notes in Computer Science(), vol 10010. Springer, Cham. https://doi.org/10.1007/978-3-319-50062-1_17
Download citation
DOI: https://doi.org/10.1007/978-3-319-50062-1_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-50061-4
Online ISBN: 978-3-319-50062-1
eBook Packages: Computer ScienceComputer Science (R0)