Università di Roma “Tor Vergata”

University of Nanjing

This is a course in algebraic number theory. The main topic of study is the theory of number fields, i.e. of finite extensions of Q and their rings of integers. We prove the two basic finiteness theorems in this theory: the ideal class group of the ring of integers of a number field is finite and the unit group of the ring of integers of a number field is finitely generated.
These two theorems are key ingredients of the proofs of some of the most important results in 20th century number theory. They play for instance fundamental roles in the proof of the famous Mordell-Weil theorem for abelian varieties and of Faltings' proof of the Mordell conjecture.
Prerequisites are basic Linear algebra and topology, basic algebra: groups, rings and fields.

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