# Number Theory Lecture 2

This lecture was given on September 1st 2009.

A (comm.) ring is a Dedekind domain if it is

- Noetherian
- a domain
- Integrally closed
- every nonzero prime ideal is maximal

Prop: Let be an integral domain, . Let be a finite extension. Let be the integral closure of in . Then is a subring of , and is integrally closed in .

the first two claims should be done as exercises, the third is more involved and found in many comm. alg. texts.

Prop: With the same notation as above, suppose $latexR$ is Dedekind. Assume either characteric 0, or simply that L/K is seperable. Then is Dedekind and is a finite -module whose span is .

Rmk: If the definion of Dedekind required to be a PID, then the above proposition would be false. The definition of a Dedekind domain is such that it is perserved under finite integral closure.

Let be a number field ( a finite extenion of the rationals). The ring of integers of , denoted is the integral closure of the integers in .

Cor: is a dedekind domain and a finite free module of rank equal to .

proof: the first claim follows from the prev prop, for the second claim use the structure theorem for abelian groups.

Rmk: If is a finite extension of number fields then is equal to the integral closure of in .

Rmk: For a Dedekind domain, being a PID is equivalent to being a UFD.

### Fractional Ideals

Summary of rest of lecture. Prop is a dedekind domain iff all fractional ideals are invertible. Thus let denote the abelian group under ideal multiplication.

Thm: is free abelian generated by the non-zero primes.

For there is a natuaral map sending to the principal ideal it genreates . So there is an injection , define to be the cokernel of this injection. In otherwords, there is an exact squence

Thm: For a number field , is finite.

Rmk: This construction is analogous to the Picard group of an algebraic curve. Namely from the exact sequence of sheaves on a curve :

one gets from the long exact sequence on cohomology

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- September 2, 2009 / 12:33 pm

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