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Integer-Valued Polynomials over Quaternion Rings
Werner, Nicholas J.

2010, Doctor of Philosophy, Ohio State University, Mathematics.
When D is an integral domain with field of fractions K, the ring Int(D) of integer-valued polynomials over D is defined to be the set of all polynomials f(a) in K[x] such that f(a) is in D for all a in D. The goal of this dissertation is to extend the integer-valued polynomial construction to certain noncommutative rings. Specifically, for any ring R, we define the R-algebra RQ to be the set of elements of the form a + bi + cj + dk, where i, j, and k are the standard quaternion units satisfying the relations i2 = j2 = -1 and ij = k = -ji. When this is done with the integers ℤ, we obtain a noncommutative ring ℤQ; when this is done with the rational numbers ℚ, we get a division ring ℚQ. Our main focus is on the construction and study of Int(ℤQ), the set of integer-valued polynomials over ℤQ. We also consider Int(R), where R is an overring of ℤQ in ℚQ. In this treatise, we prove that for such an R, Int(R) has a ring structure and investigate elements, generating sets, and prime ideals of Int(R). The final chapter examines the idea of integer-valued polynomials on subsets of ℤQ.
K. Alan Loper, PhD (Advisor)
S. Tariq Rizvi, PhD (Committee Member)
Daniel Shapiro, PhD (Committee Member)

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Werner, N. (2010). Integer-Valued Polynomials over Quaternion Rings. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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Werner, Nicholas. "Integer-Valued Polynomials over Quaternion Rings." Electronic Thesis or Dissertation. Ohio State University, 2010. OhioLINK Electronic Theses and Dissertations Center. 25 Sep 2017.

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Werner, Nicholas "Integer-Valued Polynomials over Quaternion Rings." Electronic Thesis or Dissertation. Ohio State University, 2010. https://etd.ohiolink.edu/

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