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\huge{Number Theory Syllabus}\\
\large{Last verified: December 2016}
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The study of number theory inevitably includes a knowledge of the
problems and techniques of elementary number theory, however the tools
which have evolved to address such problems and their generalizations
are both analytic and algebraic, and often intertwined in surprising
ways.
The candidate should be well-versed in the concepts, definitions,
examples, theorems and proofs in elementary, analytic, and algebraic
number theory, and demonstrate facility with the techniques in
either analytic or algebraic number theory.
The candidate should easily be able to give all relevant definitions,
provide standard examples, state major theorems, and provide the ideas
behind their proofs. Specific topics which will be covered include:
\subsection*{Elementary Number Theory:}
\begin{compactenum}
\item Unique factorization theory in $\mathbb{Z}$ and applications
\item Congruences
\item Primitive roots
\item Quadratic reciprocity
\item Gauss sums
\item Arithmetic functions and Mobius inversion
\item Average values of arithmetic functions
\end{compactenum}
\subsection*{Analytic Number Theory:}
\begin{compactenum}
\item Dirichlet series and Euler products
\item Characters on abelian groups
\item The Riemann zeta function
\begin{compactenum}
\item Poisson summation, analytic continuation, and the functional
equation for the zeta function
\item Zeros (trivial and nontrivial) of the zeta function, Riemann hypothesis
\end{compactenum}
\item Dirichlet $L$-series, prime number theorem, distribution of prime numbers
\item Dirichlet's theorem on primes in arithmetic progressions
\item Prime number theorem for arithmetic progressions
\end{compactenum}
\vfill
\rightline{(continued on reverse)}
\subsection*{Algebraic Number Theory:}
\subsubsection*{Global Theory:}
\begin{compactenum}
\item Dedekind domains, ring of integers in a number field
\item Integral basis, fractional ideals, residue class field, norm of
an ideal
\item Ideal class group and class number
\item Minkowski's theorem on convex regions; Dirichlet's unit theorem
\item Ramification and inertial degrees, discriminant and different
\item Decomposition of prime ideals in
\begin{compactenum}
\item Arbitrary extensions of number fields
\item Galois extensions of number fields
\item Abelian extensions of number fields
\end{compactenum}
\item Decomposition and inertia groups and fields
\item Frobenius automorphism, Artin symbol
\item Kronecker-Weber theorem
\item Examples: quadratic and cyclotomic fields
\end{compactenum}
\subsubsection*{Local Theory:}
\begin{compactenum}
\item Valuations, approximation theorem for valuations
\item Completions of number fields
\item Local fields and Hensel's lemma
\item Examples: $p$-adic numbers
\end{compactenum}
\vfill
\subsection*{References:}
\newcommand{\reference}[2]{\item #1, \textit{#2} \par}
\begin{compactitem}
\reference {Apostol}{Introduction to Analytic Number Theory}
\reference {Davenport}{Multiplicative Number Theory}
\reference {Ireland, Rosen}{A Classical Introduction to Modern Number
Theory}
\reference {Janusz}{Algebraic Number Fields}
\reference {Marcus}{Number Fields}
\reference {Ribenboim}{Classical Theory of Algebraic Numbers}
\end{compactitem}
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