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Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers , finite fields , and function fields .
Algebraic number theory studies the arithmetic of algebraic number fields — the ring of integers in the number field, the ideals and units in the ring of integers, the extent to which unique factorization holds, and so on.
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Algebraic number theory is the study of roots of polynomials with rational or integral coefficients. These numbers lie in algebraic structures with many similar properties to those of the integers.
Course Description. This is the first semester of a one-year graduate course in number theory covering standard topics in algebraic and analytic number theory. At various points in the course, we will make reference to material from other branches of mathematics, including topology, complex analysis, representation theory, and algebraic ….
Algebraic Number Theory. Ben Green. Contents. Preface. 0.1. A brief introduction. 0.2. These notes. 0.3. Prerequisites. Chapter 1. Algebraic numbers. 1.1. Algebraic numbers. Minimal polynomials. 1.2. The algebraic numbers are a eld. 1.3. Number elds. The primitive element theorem. 1.4. More examples. 1.5. Conjugates and embeddings. 1.6. Norms. 1.7.
A large part of the motivation for algebraic number theory comes from trying to solve Dio-phantine equations. In other words, if we take a polynomial p(x) ∈Z[x 1,...,x n], we can ask which values a= (a 1,...,a n) ∈Zn satisfy p(a) = 0. This is an extremely difficult question in general, as illustrated by the following theorem.
Algebraic number theory is a subject that came into being through the attempts of mathe-maticians to try to prove Fermat’s last theorem and that now has a wealth of applications to Diophantine equations, cryptography, factoring, primality testing, and public-key cryp-tosystems.
