Number Theory Lecture 2

Lecture2

This lecture was given on September 1st 2009.

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

1. Noetherian
2. a domain
3. Integrally closed
4. every nonzero prime ideal is maximal

Prop: Let $R$ be an integral domain, $K = Frac(R)$.  Let $L/K$ be a finite extension.  Let $S$ be the integral closure of $R$ in $L$.  Then $S$ is a subring of $L$, $Frac(S) = L$ and $S$ is integrally closed in $L$.

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 $S$ is Dedekind and $S$ is a finite $R$-module whose $K$ span is $L$.

Rmk: If the definion of Dedekind required $R$ 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 $E/\mathbb{Q}$ be a number field ( a finite extenion of the rationals).  The ring of integers of $E$, denoted $O_E$ is the integral closure of the integers in $E$.

Cor: $O_E$ is a dedekind domain and a finite free $\mathbb{Z}$ module of rank equal to $[E:\mathbb{Q}]$.

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

Rmk: If $E/F$ is a finite extension of number fields then $O_E$ is equal to the integral closure of $O_F$ in $E$.

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

Fractional Ideals

Summary of rest of lecture.  Prop $R$ is a dedekind domain iff all fractional ideals are invertible.  Thus let $\mathscr{J}(R)$ denote the abelian group under ideal multiplication.

Thm: $\mathscr{J}(R)$ is free abelian generated by the non-zero primes.

For $K = frac(R)$ there is a natuaral map $K^* \to \mathscr{J}(R)$ sending $x$ to the principal ideal it genreates $(x)$.  So there is an injection $0 \to K^*/R^* \to \mathscr{J}(R)$, define $CL(R)$ to be the cokernel of this injection. In otherwords, there is an exact squence

$0 \to R^* \to K^* \to \mathscr{J}(R) \to CL(R) \to 0$

Thm: For a number field $E$, $CL(O_E)$ 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 $X$ :

$0\to O^*\to K^* \to K^*/O^*\to 0$

one gets from the long exact sequence on cohomology

$0 \to \Gamma(O^*) \to \Gamma(K^*) \to Cart(X) \to Pic(X) \to 0$