**2006.06.30:** End of proof of Dirichlet's unit theorem.

**2006.06.27:** End of proof of theorem 37. Misprint on p. 140: In the expression for V_{0,s}(1) at the bottom of page 140, a factor V_{0,s-1}(1) is missing from second equality sign on.

Begin proof of Dirichlet's unit theorem; this is theorem 38, p. 142.

We will need this note:

Discrete subgroups, lattices, and units.

This note will *replace* the following: From the beginning of the proof of theorem 38, p. 142, until and including lemma 2, p. 145.

**2006.06.23:** Quiz (90 minutes).

**2006.06.20:** Chap. 5 continued: Minkowski's lemma on convex bodies (lemma on p. 137) and the important theorem 37.

Part (i) of Proposition 1 of the following note is implicit in a couple of places, - notably so in the proof of Minkowski's lemma in the 'compact case':

Discrete subgroups, lattices, and units.

**2006.06.14:** Chap. 5 continued. Ready to prove Minkowski's lemma on convex bodies, - this is the lemma on p. 137.

**2006.06.09: **End chap. 3: End of proof of theorem 25.

Prime decomposition in cyclotomic fields. I will prove theorem 26 only in the case where m is a prime power.

We skip theorem 27.

Misprint on p. 77: The p^2 around the middle of the page should read p^a. (But if you take notes at the lecture you don't have to read the proof in the book anyway).

We began chap. 5: Minkowski's geometry of numbers, finiteness of the class number, and Dirichlet's unit theorem.

**2006.06.06: **Chap. 3 continued: Prime decomposition in quadratic number fields. We began the proof of theorem 25.

**2006.06.02:** Chap. 3 continued: Galois action on prime decompositions.

Ramification: Theorem 24.

Misprint on p. 73: At the end of line 2 we should have $\sigma^{-1}(Q)$ and not $\sigma^{-1}(\alpha)$.

**2006.05.30: **Chap. 3 continued. Ready to prove theorem 23.

**2006.05.26: **Chap. 3 continued.** ** We started the proofs of theorems 21 and 22.

**2006.05.23: **Chap. 3 continued.** ** Until Th. 20, p. 63.

We skip theorems 17 and 18, at least for now.

**2006.05.19: **Chap. 3 continued. Until Cor. 3 on p. 59.

**2006.05.16:** End chap.2: Integral bases in cyclotomic fields. We leave chap. 2 after the proof of Theorem 11.

Begin chap. 3: Dedekind domains: We started the proof of Theorem 14.

**2006.05.12:** Holiday, - no lectures.

**2006.05.09:** Chap. 2 continued: Integral bases.

**2006.05.05:** Chap. 2 continued: Discriminants.

Note about the Vandermonde determinant:

http://www.math.ku.dk/~kiming/lecture_notes/2001-2002-mat1gb/vandermonde.pdf

**2006.05.02:** Chap. 2 continued: Algebraic integers, trace and norm.

Concerning the definition at the bottom of p. 16: If K is an algebraic number field the set of algebraic integers in K forms a subring of K. The book tried to introduce the new name 'number ring of K' for this ring but this has not become standard notation. Instead this ring is referred to as the 'ring of algebraic integers in K'. It is often denoted by the symbol O_K.

**2006.04.28:** Begin chap. 2: Algebraic integers.

I find the following variant of Theorem 1 (p. 14) more natural: A number is an algebraic integer if and only if its minimal polynomial has integer coefficients (recall that the minimal polynomial of an algebraic number \alpha is the monic polynomial of smallest degree with rational coefficients in which \alpha is root).

The proof of this statement is again an immediate consequence of the lemma on p. 14.

With this alternative formulation of theorem 1, the proofs of corollaries 1 and 2 become clearer I think ...

**2006.04.25:** Introduction and motivation: Chap. 1.