Number Fields(D)
NUMBER FIELDS (D)
Part IB Groups, Rings and Modules is essential and Part II Galois Theory is desirable.
- Definition of algebraic number fields, their integers and units. Norms, bases and discriminants. [3]
- Ideals, principal and prime ideals, unique factorisation. Norms of ideals. [3]
- Minkowski’s theorem on convex bodies. Statement of Dirichlet’s unit theorem. Determination of units in quadratic fields. [2]
- Ideal classes, finiteness of the class group. Calculation of class numbers using statement of the Minkowski bound. [3]
- Dedekind’s theorem on the factorisation of primes. Application to quadratic fields. [2]
- Discussion of the cyclotomic field and the Fermat equation or some other topic chosen by the lecturer. [3]
Appropriate books
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- Alan Baker A Comprehensive Course in Number Theory. Cambridge University Press 2012
- † Z.I. Borevich and I.R. Shafarevich Number Theory. Elsevier 1986
- † J. Esmonde and M.R. Murty Problems in Algebraic Number Theory. Springer 1999 E. Hecke Lectures on the Theory of Algebraic Numbers. Springer 1981
- † D.A. Marcus Number Fields. Springer 1977
- I.N. Stewart and D.O. Tall Algebraic Number Theory and Fermat’s Last Theorem. A K Peters 2002
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