Algebraic Number Theory.

Book: H. Koch: Number Theory. Algebraic Numbers and Functions.

Graduate Studies in Mathematics 24, American Mathematical Society 2000.

Course plan: App. B; Chap. 2.1-2.10, 2.12-2.14; Chap. 9.1-9.5; Chap. 3.1-3.3, 3.5-3.6, 3.8-3.9; + the various notes below.

12.18:  3.9 (Computing class groups): Some examples.

As another illustration of the power of the ideal theory I will also - following Dedekind - show how one can determine the rings of integers in pure cubic fields (these are fields of form $Q(\sqrt[3]{d})$). I've written up a note about it here:  pdf

Discussion of Fermat will be postponed to the continuation of the course in spring. If you are curious about Fermat but will not take the continuation, you can easily, i.e., with the knowledge that we have by now, read about it on your own. It's in chapters 6.4-6.5 of the book.

12.15: End 3.8 (Computing the decomposition of primes) + Begin 3.9 (Computing class groups).

I also spend a little time indicating the reasons why finding laws' for the decomposition of primes in algebraic number fields is one of the main motivating and deep problems of algebraic number theory. This vague question leads straight to the heart of modern number theory, more precisely the so-called Langlands program. To get in that directing, many more preparations are necessary; among a lot of other stuff one needs to learn about topics such as valuations and completions of algebraic number fields, and about the so-called L-functions. This will be a core content in the continuation of the course next spring.

12.11:  3.8 (Computing the decomposition of primes) continued.

12.08:  3.6 (Generalized congruence arithmetic) + Begin 3.8 (Computing the decomposition of primes).

The proof of Prop. 3.6.5 about the \Phi-function (generalization of Euler's \phi-function) is left as an exercise for the reader. You can try it yourself before the lecture. In the lecture I will of course give the proof; I have written a note about it:  pdf

Since we are skipping section 3.7 (localization) we will prove the first main theorem of section 3.8 (Theorem 3.8.2) without using localization. I've written a note that will replace section 3.8 until (but not including) Prop. 3.8.3. Download here:  pdf

12.04:  End 3.3 (Consequences of the main theorem) + 3.5 (Norms of ideals).

In the proof of Prop. 3.3.5 I find it easiest to use the formulas below for the prime factorization of gcd(...) and lcm(...). Notice the following: If \alpha is in A then v_P(\alpha) \ge v_P(A). On the other hand, if \alpha is not divisible by AP_i then v_{P_i}(\alpha) \le v_{P_i}(A) (otherwise one would have v_P(\alpha) \ge v_P(AP_i) for all P, and \alpha would be divisible by AP_i). So, we have v_P(\alpha) = v_P(A) whenever P is a prime divisor of B. Then we easily compute that the prime factorization of AB+O\alpha = gcd(AB,O\alpha) is the same as the prime factorization of A, i.e., AB+O\alpha = A. Similarly, by considering lcm one finds AB \cap O\alpha = \alpha B.

Misprint: On page 74: For us the ring \Gamma is Z. On line -6 on p. 74 it is said that the matrix A has coefficients in \Gamma; this is not true, we have only the coefficients in Q (that would correspond to the quotient field of \Gamma in the general case dealt with in the book). The reason is that we are talking about a general fractional ideal; then all we can say is that A has coefficients in Q (because A is a base change matrix for a linear map of the Q-vector space K into itself).

Misprint: On page 75: In the proof of Lemma 3.5.2 the 3'rd line should end with [C:CB] (and not [C:CA]).

12.01:  3.3 (Consequences of the main theorem) continued.

It's good to remember in this section that for the case O = O_K, the ring of integers of an algebraic number field K, the notion of a non-zero fractional ideal of O is the same as a complete module with order O_K: If m is a complete module with order O_K then m is an O_K-module; as such it must be finitely generated, since it is already finitely generated as a Z-module. On the other hand, if A is a non-zero fractional ideal then by definition A is finitely generated as an O_K-module. Since O_K is finitely generated as a Z-module, we see that A is finitely generated as a Z-module. So, A is a module in K. Its order must be O_K since O_K \cdot A = A (i.e., O_K is contained in the order of A as module; on the other hand this order is contained in O_K). If \alpha \neq 0 is in A, and if \omega_1, ..., \omega_n is a Z-basis of O_K (thus n = [K:Q]) we have that A contains the elements \alpha\omega_1, ..., \alpha\omega_n; these elements are however independent over Z, as the \omega's are Z-independent. It follows that A has rank at least n as a Z-module; thus the rank is exactly equal to n since we know that the rank over Z of any module in K is at the most n. Thus, A is a complete module with order O_K. This then has the following immediate consequence: The class group - defined as the group I_O of non-zero fractional ideals modulo the subgroup of principal ideals, i.e., fractional ideals of the form O_K \cdot \alpha with \alpha \neq 0 in K - is in this case nothing but the set of representatives of complete modules with order O_K w.r.t. the equivalence relation that we have introduced much earlier. In particular, the class group in this case is a finite group (by our earlier theorem on the finiteness of class numbers of algebraic number fields).

I found it a bit more natural to prove Prop. 3.3.3 before Prop. 3.3.2. The point is of course that Prop. 3.3.3 can be used nicely in the proof of Prop. 3.3.2 (in fact is implicitly more or less needed).

On p. 72: Instead of the references to App. A.1, we use the following direct definitions of gcd and lcm of ideals: Let A and B be ideals. Then gcd(A,B) is the smallest ideal dividing both A and B, i.e., the smallest ideal containing both A and B, i.e., gcd(A,B) = A+B. Similarly, lcm(A,B) is defined as the largest ideal divisible by both A and B; this is the largest ideal contained in both A and B, i.e., lcm(A,B) = A \cap B.

I then showed that one has the usual formulas for the prime factorization of the ideals gcd(A,B) and lcm(A,B): Namely that v_P(gcd(A,B)) = min{v_P(A),v_P(B)}, and v_P(lcm(A,B)) = max{v_P(A),v_P(B)} for every prime ideal P. The proof of the first is as follows: Let C be the ideal with v_P(C) = min{v_P(A),v_P(B)}for each P. Then one sees that C divides both A and B (there is an ideal D with CD = A, and similarly for B). So, C must divide gcd(A,B) which gives that v_P(gcd(A,B)) \ge v_P(C) = min{v_P(A),v_P(B)}. On the other hand, gcd(A,B) divides both A and B, so v_P(gcd(A,B)) is less than or equal to both v_P(A) and v_P(B), i.e., v_P(gcd(A,B)) \le  min{v_P(A),v_P(B)}. The desired follows. The proof of the formula for lcm is completely analogous.

11.27:  End 3.1, 3.2 (Dedekind rings and prime factorization) + Begin 3.3 (Consequences of the main theorem).

Misprint: On page 71 the sentence ... we denote the set of nonzero prime ideals of O by I_O.' should read: ... we denote the set of nonzero fractional ideals of O by I_O.'

For the proof of Lemma 3.2.5 you should imitate the proof of Prop. 2.3.2 (that we only discussed for the case where O_K is the ring of algebraic integers in an algebraic number field K). I.e., you write P = O\mu_1 + ... O\mu_s (since P is finitely generated). Then we must have \gamma \mu_i = \sum_j \alpha_{i,j} \mu_j with \alpha_{i,j} in O. As in the proof of Prop. 2.3.2 this implies that \gamma is root of a monic polynomial with coefficients in O. As O is integrally closed this gives \gamma \in O, and contradiction.

11.24:  3.1, 3.2 (Dedekind rings and prime factorization) continued.

11.20:  End 9.5 (Fundamental units in real quadratic orders) + Begin 3.1, 3.2 (Dedekind rings and prime factorization).

At the beginning of Chap. 3 the book refers to Appendix A.1. It's not necessary to read that: All you need to know is that an irreducible element \pi of O_K by definition is an element that is not a unit and has the property that if \pi = \alpha \beta with \alpha,\beta \in O_K then 1 of \alpha, \beta is a unit. In the first example of Chap. 3 one then verifies easily by looking at the norms that 3, 7, and 1+2\sqrt{-5} are all irreducible elements of Q(\sqrt{-5}). Thus they give an example of non-unique factorization into products of irreducibles.

Also, remember that indices are just natural numbers for us (in contrast with the book which in an appendix interprets indices as ideals). Thus for instance, (3.1.2) in the proof of Theorem 3.1.3 should be read as [O:A_1] \ge [O:A_2] \ge ... , as I explained in the lecture. These inequalities follow immediately from A_1 \subseteq A_2 \subseteq ..., and is equivalent to (3.1.2) in the book when you interpret indices as ideals (the 'inequalities' become inverted then).

11.17:  9.5 (Fundamental units in real quadratic orders) continued.

We discuss the famous continued fractions algorithm for solving the problem of determining a fundamental unit in an arbitrary order of a real quadratic number field. This problem can also be formulated as a problem of finding solutions to Pell equations. There is in Notices of the American Mathematical Society a very nice article by Lenstra on Pell equations, continued fractions, and other stuff, and also about the connection of Archimedes cattle problem' to these other questions. You'll see why this was a hard problem in antiquity! The link to the article is here.

In the proof of Theorem 9.5.5 it is of course not so good to write a'_i for the numbers in the continued fraction expansion of \theta (since we use x' to denote the conjugate of x in the fixed quadratic field).

Misprint on the second line of p. 299: \frac{p}{q} \le 1 should be \frac{p}{q} \ge 1.

In the proof of Theorem 9.5.2 it is of course tacitly assumed that \epsilon is not \pm 1.

11.13:  End 9.3 + 9.4 (Continued fractions) + Begin 9.5 (Fundamental units in real quadratic orders).

Misprint on the last line of p. 294: \theta'_{l-1+k} should be \theta'_{n-1+k}.

11.10:  Further remarks about class numbers (without proofs).  Begin 9.3 + 9.4 (Continued fractions).

The results in section 9.3 are elementary and the proofs are easy. I will explain the results but the proofs will be left to read on your own. I will give all proofs in sections 9.4 and 9.5.

11.06:  9.2 (Class numbers in the imaginary quadratic case).

The proof of theorem 9.1.9 is perhaps a bit concentrated. I've written up a note with all details:  pdf

10.30: 9.1 (Quadratic forms and orders in quadratic number fields) (partly seminar by Henrik Friis Christensen).

10.27:  End 2.13 (Geometry of numbers) + Begin 9.1 (Quadratic forms and orders in quadratic number fields). Seminar by Henrik Friis Christensen.

There is a misprint at the line 3 of page 277: The condition should read: $a \neq 0$ and $c \neq 0$.

10.23:  2.13 (Geometry of numbers) continued.

Concerning the inequality between arithmetic and geometric means: If you have not seen this before, you can find a very short proof in the article: Horst Alzer: A Proof of the Arithmetic Mean-Geometric Mean Inequality', The American Mathematical Monthly, Vol. 103, No. 7. (Aug. - Sep., 1996), p. 585. We have online access to this article via www.jstor.org . For another proof look at this page.

The proof of Hermite's Discriminant Theorem is clear, but not crystal clear: there's a slight problem with the argument at the end if $r_1 = 1$. But it's easy to repair this small inaccuracy: One just have to redefine slightly the set $M$ occurring in the proof:  pdf

10.20:  End 2.12 (Lattices) + Begin 2.13 (Geometry of numbers).

10.13, 10.17:  Autumn vacation.

10.09:  2.10 (Proof of Dirichlet's unit theorem continued) + Begin 2.12 (Lattices).

There is an error in the beginning of section 2.12: The definition of a lattice in $R^n$ is that it is a subgroup of the form $Zv_1 + \ldots + Zv_n$ where$v_1, \ldots , v_n$ is a basis of $R^n$ (as a real vector space). So, a lattice is a finitely generated subgroup of $R^n$ of rank $n$; the converse is however not true; but it is true if one additionally assumes that the subgroup is discrete in $R^n$: See this note (which also contains some remarks about the beginning of section 2.13):  pdf

I want to give 2 slightly differing versions of Minkowski's lattice point theorem, and reshape the proof to make it a bit clearer.

10.06:  End 2.9 + 2.10 (Proof of Dirichlet's unit theorem continued).

10.02: Begin 2.9 (Proof of Dirichlet's unit theorem).

09.29:  End 2.14 (Application to norm form equations) + 2.8 (Units).

Warning:  Theorem 2.14.2 is wrong as stated. The correct formulation is that equation (2.14.2) gives a surjective map from classes of modules to classes of complete forms of degree $n$ that split into linear factors in $K$. However, this does not in general result in a 1-1 correspondence, i.e., the map is not necessarily injective. I.e., the problem is that one can have 2 inequivalent modules that give rise to the same form. This may happen for instance in the case where $K/Q$ is Galois: One can construct examples of inequivalent modules that are conjugate, i.e., one is obtained from the other by acting with an element of the Galois group; but conjugate modules obviously give rise to the same decomposable form.

09.25:  End 2.7 (Class number) + Begin 2.14 (Application to norm form equations).

In connection with 2.14 I'm also starting the discussion about units. In connection with that and with 2.14, I have found it convenient to make explicit a small observation that is only implicitly noted in the book. Have a look at this note:  pdf  We also made a useful observation that occurs as part of Lemma 2.10.2 of the book: I have updated the note to include this argument.

09.22:  End 2.5 (integers in quadratic fields) + 2.6 (Examples of ring of integers) + Begin 2.7 (Class number).

In connection with section 2.6 I said a little about the program GP/PARI which is a package for computations in algebraic number theory. You can start the program on our system by issuing the command gp' (without quotes). The program can also be downloaded for free at http://pari.math.u-bordeaux.fr/ (there are versions for various platforms available).

Note to section 2.7: In the book it looks as if the proof of Theorem 2.7.1 ends on page 47. It doesn't: The argument continues after the remark on page 48.

For the proof of equation (2.7.4) we will use the argument in this note:  pdf

09.18:  End 2.4 + Begin 2.5 (integers in quadratic fields).

09.15:  Final remarks about App. B + Begin 2.4 (Complete modules).

We will note that if $\theta$ is a primitive element of $L/K$ ($L$ of degree $n$ over $K$), then the discriminant of the basis $1,\theta,\ldots,\theta^{n-1}$ is the same as the discriminant of the minimal polynomial of $\theta$. The argument is immediate if one uses Vandermonde's determinant. For those who have never seen Vandermonde's determinant I have a note about it here (unfortunately only in Danish):  pdf

09.11:   Appendix B (Bilinear forms, trace, norm).

By a module in an algebraic number field $K$ one understands a finitely generated subgroup of the additive group of $K$, i.e., something of the form $Z\mu_1 + \ldots + Z\mu_s$ where the $\mu_i$ are numbers in $K$. Unfortunately, this definition is only between the lines on p. 41 of the book.