**Weekly
updates:**

**2007.06.29: **End of proof of Dirichlet's unit theorem.

**2007.06.26: **We formulate Dirichlet's unit
theorem and begin the proof of it; this is theorem 38, p. 142.

We will need this note:

Discrete subgroups, lattices, and logarithmic embedding of 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.

**2007.06.22:** Chap. 5 continued: We finished the proofs of
Minkowski's lemma and 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.

**2007.06.19:** Chap. 5 continued: First we prove further
corollaries to theorem 37, and then we proceed to proofs of 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 logarithmic embedding of units.

Ww finished the proof of part (i) of the proposition.

**2007.06.15:** Chap. 5 continued. We discussed the
embedding of an algebraic number field of degree n into R^n, defined and
discussed the notion of a lattice in R^n, proved theorem 36, formulated theorem
37 and proved corollary 1 to theorem 37.

**2007.06.13: **End chap. 3: Prime decomposition in
cyclotomic fields.

This is theorem 26 which we prove only in the case where m is a prime power. Here is a note with the proof I shall give for that case:

Prime decomposition in cyclotomic fields.

If you also want additionally to read the proof in the book then notice a misprint on p. 77: The p^2 around the middle of the page should read p^a.

After this we skip theorem 27 and begin Chap. 5: Minkowski's geometry of numbers, finiteness of the class number, and Dirichlet's unit theorem.

We recalled the definition of class number and class group for algebraic number fields, and reviewed the application of the regularity condition for primes to the first case of Fermat's last theorem.

**2007.06.11:** Chap. 3 continued: Corollaries to theorem 24.

After that we proved and discussed theorem 25 which concerns prime decomposition in quadratic number fields.

**2007.06.08: **Chap. 3 continued: Galois action on prime
decompositions and ramification: theorems 23 and 24. Until and
including corollary 1 to theorem 24.

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

**2007.06.01: **Chap. 3 continued: We finished the proofs
of theorems 21 and 22.

**2007.05.29:** Chap. 3 continued. We started the proofs of
theorems 21 and 22.

Theorem 21 is a fundamental theorem in algebraic number theory.

**2007.05.25:** Chap. 3 continued. Until and
including theorem 20, p. 63.

We skip theorems 17 and 18.

We also discussed the notions of `greatest common divisor' and `least common multiple' for ideals in Dedekind domains. It was an easy exercise to determine the prime factorizations of gcd(A,B) and of lcm(A,B) from the prime factorizations of A and B.

**2007.05.22:** Chap. 3 continued. Until and
including corollary 3 on p. 59.

**2007.05.18:** End chap.2: Integral bases in cyclotomic fields:
Theorem 10. We leave chap. 2 after theorem 11.

Begin chap. 3: Dedekind domains. We began the proof of theorem 14.

**2007.05.15:** Chap. 2 continued: Integral bases, integral
bases in cyclotomic fields. We proved theorem 11 and set up everything for
the proof of theorem 10.

**2007.05.11:** Chap. 2 continued: Discriminants,
discriminants of cyclotomic fields, structure of the ring of algebraic integers
in a number field.

We used the result of problem 16, p. 9, which shows that 1-\omega has norm p if \omega is a primitive p'th root of unity in the p'th cyclotomic field (p odd prime).

We discussed the fact that if \alpha is an arbitrary algebraic number then there is a natural number m such that m\alpha is an algebraic integer. A consequence of this is that an algebraic number field K is always the field of fractions of its ring of algebraic integers O_K, and that we always have a Q-basis of K consisting of elements of O_K.

We set up everything so that we are ready to prove theorem 9.

Here is a note about the Vandermonde determinant, in case you are not familiar with it:

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

**2007.05.08:** Chap. 2 continued: Trace and norm,
discriminants.

I will suppose that you are familiar with the elementary properties of cyclotomic fields, as given in the section `The cyclotomic fields', pp. 17-19 in the book.

If you are curious about the connection between disciminants and bilinear forms that I mentioned briefly, here is a note about it (but I won't discuss this further, and won't use it):

**2007.04.27:** 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 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.

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.

**2007.04.24:** Introduction and motivation: Chap.
1.