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\title{Number Theory}
\author{Last Updated: August 1983}
\date{}

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The candidate will be expected to have a command of two of the three
branches of number theory (elementary, algebraic and analytic) and
have an acquaintance with the basic concepts, techniques and major
theorems of the third.

\subsection*{Elementary Number Theory:}

\begin{enumerate}
\item Unique factorization theory in $\mathbb{Z}$
\item Congruences
\item Primitive roots
\item Quadratic reciprocity
\item Gauss sums
\item Number-theoretic functions and Mobius inversion
\item The distribution of primes
\item One of Diophantine equations or the theory of partitions
\end{enumerate}

\subsection*{Analytic Number Theory:}

\begin{enumerate}
\item Dirichlet series and Euler products
\item Characters on abelian groups
\item The Riemann zeta function
  \begin{enumerate}
  \item Poisson summation, analytic continuation, and the functional
    equation for the zeta function
  \item Zeros of the zeta function, trivial, non-trivial, Riemann
    Hypothesis
  \end{enumerate}
\item  $L$-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{enumerate}

\subsection*{Algebraic Number Theory:}

\subsubsection*{Global Theory:}

\begin{enumerate}
\item Dedekind domains
\item 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
\item Dirichlet's Unit theorem
\item Decomposition of prime ideals in
  \begin{enumerate}
  \item arbitrary extensions of number fields
  \item Galois extensions of number fields
  \item Abelian extensions of number fields
  \end{enumerate}
\item Ramification and inertial degrees, discriminant and different
\item Decomposition and Inertia groups and fields
\item Frobenius automorphism, Artin symbol
\item Kronecker-Weber theorem
\item Examples: Quadratic and Cyclotomic fields
\end{enumerate}

\subsubsection*{Local Theory:}

\begin{enumerate}
\item Valuations
\item Approximation theorem for valuations
\item Completions of number fields
\item Local fields and Hensel's lemma
\item Examples: $p$-adic numbers
\end{enumerate}

\subsection*{References:}

\subsubsection*{Elementary Number Theory}

\begin{enumerate}
\item Adams, Goldstein: \textit{Introduction to Number Theory}
\item Apostol: \textit{Introduction to Analytic Number Theory}
\item Ireland, Rosen: \textit{A Classical Introduction to Modern Number
  Theory}
\item LeVeque: \textit{Topics in Number Theory}
\item Niven, Zuckerman: \textit{Number Theory}
\end{enumerate}

\subsubsection*{Analytic Number Theory}

\begin{enumerate}
\item Apostol: \textit{Introduction to Analytic Number Theory}
\item Chandresekharan: \textit{Introduction to Analytic Number Theory}
\item Rademacher: \textit{Topics in Analytic Number Theory}
\item Titchmarsh: \textit{Theory of the Riemann Zeta Function}
\end{enumerate}

\subsubsection*{Algebraic Number Theory}

(G = Global; L = Local)
\begin{enumerate}
\item (L) Artin: \textit{Algebraic Numbers and Algebraic Functions}
\item (G,L) Cassels, Frohlich: \textit{Algebraic Number Theory}
\item (G,L) Golstein: \textit{Analytic Number Theory (Chapters 1-6)}
\item (G,L) Janusz: \textit{Algebraic Number Fields}
\item (G) Lang: \textit{Algebraic Number Theory}
\item (G) Marcus: \textit{Number Fields}
\item (G) Ribenboim \textit{Algebraic Numbers}
\item (L) Weiss: \textit{Algebraic Number Theory}
\end{enumerate}

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