Some applications of algebraic number theory

The following examples illustrate that learning algebraic number theory as soon as possible is an excellent investment of your time.

1. Integer factorization using the number field sieve. The number field sieve is the asymptotically fastest known algorithm for factoring general large integers (that don't have too special of a form). Recently, in December 2003, the number field sieve was used to factor the RSA-576 $10000 challenge:
   $$\begin{array}{l} 1881988129206079638386972394616504398071635... ...hspace{3em}\ldots6459852171130520711256363590397527 \end{array}$$
   (The $\ldots$ indicates that the newline should be removed, not that there are missing digits.)

2. Primality test: Agrawal and his students Saxena and Kayal from India found in 2002 the first ever deterministic polynomial-time (in the number of digits) primality test. There methods involve arithmetic in quotients of $(\mathbf{Z}/n\mathbf{Z})[x]$, which are best understood in the context of algebraic number theory. For example, Lenstra, Bernstein, and others have done that and improved the algorithm significantly.

3. Deeper point of view on questions in number theory:
   1. Pell's Equation ( $x^2-dy^2=1$) $\Longrightarrow$ Units in real quadratic fields $\Longrightarrow$ Unit groups in number fields
   2. Diophantine Equations $\Longrightarrow$ For which $n$ does $x^n+y^n=z^n$ have a nontrivial solution?
   3. Integer Factorization $\Longrightarrow$ Factorization of ideals
   4. Riemann Hypothesis $\Longrightarrow$ Generalized Riemann Hypothesis
   5. Deeper proof of Gauss's quadratic reciprocity law in terms of arithmetic of cyclotomic fields $\mathbf{Q}(e^{2\pi i/n})$, which leads to class field theory.
4. Wiles's proof of Fermat's Last Theorem, i.e., that the equation $x^n+y^n=z^n$ has no solutions with $x,y,z,n$ all positive integers and $n\geq 3$, uses methods from algebraic number theory extensively, in addition to many other deep techniques. Attempts to prove Fermat's Last Theorem long ago were hugely influential in the development of algebraic number theory by Dedekind, Hilbert, Kummer, Kronecker, and others.
5. Arithmetic geometry: This is a huge field that studies solutions to polynomial equations that lie in arithmetically interesting rings, such as the integers or number fields. A famous major triumph of arithmetic geometry is Faltings's proof of Mordell's Conjecture.
   Theorem 1.3.1 (Faltings)   Let $X$ be a nonsingular plane algebraic curve over a number field $K$. Assume that the manifold $X(\mathbf{C})$ of complex solutions to $X$ has genus at least $2$ (i.e., $X(\mathbf{C})$ is topologically a donut with two holes). Then the set $X(K)$ of points on $X$ with coordinates in $K$ is finite.
   For example, Theorem 1.3.1 implies that for any $n\geq 4$ and any number field $K$, there are only finitely many solutions in $K$ to $x^n+y^n=1$.

   A major open problem in arithmetic geometry is the Birch and Swinnerton-Dyer conjecture. An elliptic curves $E$ is an algebraic curve with at least one point with coordinates in $K$ such that the set of complex points $E(\mathbf{C})$ is a topological torus. The Birch and Swinnerton-Dyer conjecture gives a criterion for whether or not $E(K)$ is infinite in terms of analytic properties of the $L$-function $L(E,s)$.