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

 

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