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On the roots of polynomials modulo primes

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TitleInfo
Title
On the roots of polynomials modulo primes
Name (type = personal)
NamePart (type = family)
Bryk
NamePart (type = given)
John T.
NamePart (type = date)
1980-
DisplayForm
John Bryk
Role
RoleTerm (authority = RULIB)
author
Name (type = personal)
NamePart (type = family)
Tunnell
NamePart (type = given)
Jerrold B.
DisplayForm
Jerrold B. Tunnell
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
chair
Name (type = personal)
NamePart (type = family)
Iwaniec
NamePart (type = given)
Henryk
DisplayForm
Henryk Iwaniec
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Advisory Committee
Role
RoleTerm (authority = RULIB)
internal member
Name (type = personal)
NamePart (type = family)
Miller
NamePart (type = given)
Stephen D.
DisplayForm
Stephen D. Miller
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
internal member
Name (type = personal)
NamePart (type = family)
Buhler
NamePart (type = given)
Joe P.
DisplayForm
Joe P. Buhler
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
outside member
Name (type = corporate)
NamePart
Rutgers University
Role
RoleTerm (authority = RULIB)
degree grantor
Name (type = corporate)
NamePart
Graduate School - New Brunswick
Role
RoleTerm (authority = RULIB)
school
TypeOfResource
Text
Genre (authority = marcgt)
theses
OriginInfo
DateCreated (qualifier = exact)
2012
DateOther (qualifier = exact); (type = degree)
2012-05
Place
PlaceTerm (type = code)
xx
Language
LanguageTerm (authority = ISO639-2b); (type = code)
eng
Abstract (type = abstract)
We study the problem of counting the number of roots of an irreducible polynomial $f(X) in mathbb{Z}[X]$ modulo rational primes. We consider the family of polynomials $f_n(X) = X^n-X-1$, which have Galois groups isomorphic to $S_n$. The approach we take is to attach Galois representations to the counting problem and then to relate these to automorphic forms. In particular, we attempt to attach the representations to holomorphic forms on $GL_2$. We show this only works when $n leq 5$, and we present the solutions to the problem in the $n = 4$ and $5$ cases, following methods due to Serre, Crespo, and Buhler for explicitly constructing Galois representations. The solution to the $n = 5$ case is novel, requiring Hilbert modular forms. In solving the problem, we produce the first example of an icosahedral Hilbert form that is not the base change of a classical form.
Subject (authority = RUETD)
Topic
Mathematics
RelatedItem (type = host)
TitleInfo
Title
Rutgers University Electronic Theses and Dissertations
Identifier (type = RULIB)
ETD
Identifier
ETD_3982
PhysicalDescription
Form (authority = gmd)
electronic resource
InternetMediaType
application/pdf
InternetMediaType
text/xml
Extent
vii, 87 p. : ill.
Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references
Note (type = statement of responsibility)
by John T. Bryk
Subject (authority = ETD-LCSH)
Topic
Polynomials
Subject (authority = ETD-LCSH)
Topic
Number theory
Subject (authority = ETD-LCSH)
Topic
Galois theory
Identifier (type = hdl)
http://hdl.rutgers.edu/1782.1/rucore10001600001.ETD.000065098
RelatedItem (type = host)
TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
Identifier (type = local)
rucore19991600001
Location
PhysicalLocation (authority = marcorg); (displayLabel = Rutgers, The State University of New Jersey)
NjNbRU
Identifier (type = doi)
doi:10.7282/T3DN43M6
Genre (authority = ExL-Esploro)
ETD doctoral
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Rights

RightsDeclaration (ID = rulibRdec0006)
The author owns the copyright to this work.
RightsHolder (type = personal)
Name
FamilyName
Bryk
GivenName
John
Role
Copyright Holder
RightsEvent
Type
Permission or license
DateTime (encoding = w3cdtf); (qualifier = exact); (point = start)
2012-04-16 09:16:10
AssociatedEntity
Name
John Bryk
Role
Copyright holder
Affiliation
Rutgers University. Graduate School - New Brunswick
AssociatedObject
Type
License
Name
Author Agreement License
Detail
I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
Copyright
Status
Copyright protected
Availability
Status
Open
Reason
Permission or license
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Technical

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