RUcore Resource Object
RUcore Resource Object
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TitleOn the roots of polynomials modulo primes
NameBryk, John T. (author), Tunnell, Jerrold B. (chair), Iwaniec, Henryk (internal member), Miller, Stephen D. (internal member), Buhler, Joe P. (outside member), Rutgers University, Graduate School - New Brunswick,
Degree Date2012-05
Date Created2012
SubjectMathematics,
Polynomials,
Number theory ,
Galois theory
DescriptionWe 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.
NotePh. D.
NoteIncludes bibliographical references
Noteby John T. Bryk
Genretheses
Persistent URLhttp://hdl.rutgers.edu/1782.1/rucore10001600001.ETD.000065098
Languageeng
CollectionGraduate School - New Brunswick Electronic Theses and Dissertations
Organization NameRutgers, The State University of New Jersey
RightsThe author owns the copyright to this work.