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osu1274923342.pdf (559.94 KB)
ETD Abstract Container
Abstract Header
Integer-Valued Polynomials over Quaternion Rings
Author Info
Werner, Nicholas J.
Permalink:
http://rave.ohiolink.edu/etdc/view?acc_num=osu1274923342
Abstract Details
Year and Degree
2010, Doctor of Philosophy, Ohio State University, Mathematics.
Abstract
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
i
2
=
j
2
= -1 and
ij
=
k
= -
ji
. When this is done with the integers ℤ, we obtain a noncommutative ring ℤ
Q
; when this is done with the rational numbers ℚ, we get a division ring ℚ
Q
. Our main focus is on the construction and study of Int(ℤ
Q
), the set of integer-valued polynomials over ℤ
Q
. We also consider Int(
R
), where
R
is an overring of ℤ
Q
in ℚ
Q
. 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 ℤ
Q
.
Committee
K. Alan Loper, PhD (Advisor)
S. Tariq Rizvi, PhD (Committee Member)
Daniel Shapiro, PhD (Committee Member)
Subject Headings
Mathematics
Keywords
integer-valued polynomial
;
quaternion
;
Hurwitz quaternion
;
Hurwitz integer
Recommended Citations
Refworks
EndNote
RIS
Mendeley
Citations
Werner, N. J. (2010).
Integer-Valued Polynomials over Quaternion Rings
[Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1274923342
APA Style (7th edition)
Werner, Nicholas.
Integer-Valued Polynomials over Quaternion Rings.
2010. Ohio State University, Doctoral dissertation.
OhioLINK Electronic Theses and Dissertations Center
, http://rave.ohiolink.edu/etdc/view?acc_num=osu1274923342.
MLA Style (8th edition)
Werner, Nicholas. "Integer-Valued Polynomials over Quaternion Rings." Doctoral dissertation, Ohio State University, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=osu1274923342
Chicago Manual of Style (17th edition)
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Document number:
osu1274923342
Download Count:
1,459
Copyright Info
© 2010, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.