Algebraic number fields
Localisation, discrete valuation rings.
Integral ring extensions, Dedekind domains, unique factorisation of ideals. Action of the galois group on prime ideals.
Valuations and completions of number fields, discussion of Ostrowski's theorem, Hensel's lemma, unramified, totally ramified and tamely ramified extensions of
p-adic fields.
Discriminants and Ramification.
Cyclotomic fields, Gauss sums, quadratic reciprocity revisited.
The ideal class group, finiteness of the ideal class group, Dirichlet units theorem.
1. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2nd ed., Springer-Verlag, Berlin, 1990.
2. S. Lang, Algebraic Number Theory, Addison- Wesley, 1970.
3. D.A. Marcus, Number Fields, Springer-Verlag, Berlin, 1977.
S. R. Ghorpade, Lectures on Topics in Algebraic Number Theory (Univ. zu Kiel, Germany, Dec. 2001), Mumbai, January 2002. (Last update: May 3, 2008)