Algebraic Number Theory II.

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

Graduate Studies in Mathematics 24, American Mathematical Society 2000.

Lectures: Tuesdays 13-15 and Thursdays 11-13 in Aud. 8.

Course plan: Chap. 3.12, pp. 94-96, chap. 4, chap. 6.1, 6.2 until 6.2.2, 6.3-6.5, 6.7-6.8, + the various notes below.

05.20: Ascension Day. I.e., no lectures.

05.18: End 6.7 (Upper numeration of ramification groups).

Concerning Lemma 6.7.3: To be ultra-precise (and why not?) one has to remark that the argument given actually only shows the desired for u \ge 1. But if 0 < u <1 the desired equality is immediately verified as well: v_0 \phi(u) equals uv_1, and the sum is computed to equal uv_1 + 0.(v_0 - v_1) = uv_1.

In the proof of Theorem 6.7.1 on p. 194: The reference to Prop. 3.10.5 is just the following: We know that the number of elements in G_0 is e(K/F); we also know that this number equals e(K/L)e(L/F); and the numbers e(K/L) and e(L/F) is the number of elements in (G/H)_0 and H_0, respectively.

Read the example on p. 195 concerning higher ramification groups for cyclotomic fields on your own.

05.13:  6.7 (Upper numeration of ramification groups), continued.

Concerning the proof of Lemma 6.7.2: At the beginning of the proof it is not necessary to use Proposition 4.6.6: Instead of considering \beta as in the book just work with a prime element \pi for K; the reasoning remains the same, but becomes perhaps even clearer. Notice that the minimal polynomial F of \pi has the roots h.\pi where h runs through H = Gal(K/L); so \gamma.F has the roots g.\pi where g runs through the coset \gamma in G/H.
In (6.7.7) the number v_i denotes the number of elements of (G/H)_i rather than of G_i.
In the derivation of (6.7.8) one uses 6.2.1 again as in (6.7.7), - this time for the extensions K/F and K/L. Notice also that \nu_H(h) = \nu_G(h) if h is an element of H; this is immediately verified directly from definitions.

05.11:  End 6.8 (Kummer extensions) + Begin 6.7 (Upper numeration of ramification groups).

Remark to the properties of \phi on p. 191: I prefer to say that \phi is convex, i.e., that if x_1 \le x \le x_2 then the graph of \phi in the interval [x_1,x_2] is above' the line segment joining the points (x_1,\phi(x_1)) and (x_2,\phi(x_2)). This is equivalent to saying that the function (\phi(x) - \phi(x_1))/(x-x_1) is decreasing for x \ge x_1. This is easily verified from the definition of \phi since the function [G_0:G_x]^{-1} is decreasing with x.

05.06:  End 6.5 (First case of Fermat's last theorem) + Begin 6.8 (Kummer extensions).

Concerning the proof of Prop. 6.8.2: Misprint in the displayed equation: The product is from i=0 to n/h-1 of (x - \zeta_n^{ih} \alpha).

Concerning Propositions 6.8.3 and 6.8.5: One conveniently proves these statements simultaneously and clarifies the first part of the proof of Prop. 6.8.3 (which is not crystal clear) as follows: We first consider an arbitrary prime p of F; i.e., we do not exclude the case that p divides n. Choose a prime P of K dividing the given prime p, and let e be the ramification index of P over p. Then e^{-1}\nu_P is an extension of \nu_p to K. So we have (n/e)\nu_P(\alpha) = (1/e)\nu_P(\alpha^n) = (1/e)\nu_P(a) = \nu_p(a). If p is unramified in K we have e=1; it follows that then \nu_p(a) is necessarily divisible by n.
Secondly, one assumes that p does not divide n, and that \nu_p(a) is divisible n. One shows then that p is unramified in K as in the second part of the proof of Prop. 6.8.3.

Concerning the proof of Prop. 6.8.4: First, the fact that p is unramified under the stated hypotheses clearly also follows from the previous proposition 6.8.3. Secondly, one uses the criterion given by Theorem 3.8.2 to determine the number of primes dividing p in the given Kummer extension. We proved Theorem 3.8.2 in the previous course in case that the ground field was Q, cf. for instance this note decomposition.pdf . With the tools that we have now available, the proof in the general case goes through virtually without change. So, either accept this, or work through the details on your own.

We will not prove Theorem 6.8.6.

05.04:  6.5 (First case of Fermat's last theorem), continued.

The end of the proof of Prop. 6.5.2 is a little cryptic. Proceed as follows: The first part of the proof shows that the ideals (x+y\zeta^i) and (1-\zeta) are relatively prime, i.e., that their sum is O. Then notice that the ideal A := (x+y\zeta^i) + (x+y\zeta^j) contains both (1-\zeta)y and (1-\zeta)x: For A contains (x+y\zeta^i)-(x+y\zeta^j) = \zeta^i (1-\zeta^{j-i})y ; since j-i is not divisible by p we have that \zeta^{j-i} is a primitive p'th root of unity, and hence that (1-\zeta^{j-i}) is the unique prime divisor (1-\zeta) of p in O. Similarly, A contains (x+y\zeta^i)\zeta^j - (x+y\zeta^j)\zeta^i, and so also (\zeta^j - \zeta^i)x.

Since Zx + Zy = Z, as gcd(x,y)= 1, we now deduce that A contains (1-\zeta). Since (x+y\zeta^i) is relatively prime to (1-\zeta), we have that A contains O.

Concerning Theorem 6.5.3: The p-rank of the class group Cl(K) is defined as the minimal number of generators of the largest p-subgroup of the finite abelian group Cl(K), i.e., of a p-Sylow subgroup of Cl(K). This is the same as the dimension over F_p of the subgroup of Cl(K) consisting of elements whose p'th power is 1.

To obtain (6.5.9) use the following obvious rule for 'logarithmic derivatives': (uv)'/(uv) = u'/u + v'/v for polynomials u,v in a single variable z.

04.29:  End 6.4 (Cyclotomic fields) + Begin 6.5 (First case of Fermat's last theorem).

04.27:  6.3 (Decomposition of primes in intermediate fields) + Begin 6.4 (Cyclotomic fields).

In the proof of Proposition 6.4.2 there is no need to refer to Theorem 3.12.5. We argue as follows: First we deduce that \Delta(\zeta) is a power of \ell. Consequently, \ell is the only prime that ramifies in K = Q(\zeta), - cf. Prop. 3.8.1 + Th. 3.8.2, or the note decomposition.pdf from the previous course. By our discussion of differents and discriminants - cf. the note relative_norms.pdf - we then know that the different D of K over Q is divisible only by primes lying over \ell. But according to Prop. 6.4.1 there is only 1 such prime {\mathfrak l}. So D is a power of {\mathfrak l}. By the computation it suffices then to show that \nu_{{\mathfrak l}}(D) = \nu_{{\mathfrak l}}(Element different of (\zeta - 1)); but this is assured by the proof of Th. 6.2.1 since 1-\zeta is a prime element w.r.t. {\mathfrak l}.

Misprint on p. 183: In the sentence that starts 'In Section 3.8 we showed ...' one supposes that p and q are distinct primes, and the results of section 3.8 show that the Legendre symbol (q^*/p) is 1 if p splits in Q(\sqrt{q^*}), and -1 if p is inert in this quadratic field.

One then proceeds by showing that p splits in Q(\sqrt{q^*}) if and only if p is a square modulo q: Let P be a prime over p in Q(\zeta_q), and let p_0 be the prime of Q(\sqrt{q^*}) \subseteq Q(\zeta) 'under' P. Since the Frobenius elements Fr_P and Fr_{p_0} are uniquely characterized by Fr_P(\alpha) \equiv \alpha^p (mod P) for all \alpha in O_{Q(\zeta)}, and similarly for Fr_{p_0}, we see that Fr_{p_0} is simply the restriction of Fr_P to Q(\sqrt{q^*}). Now, p splits in Q(\sqrt{q^*}) if and only if Fr_{p_0} is trivial (why?). It follows that p splits in Q(\sqrt{q^*}) if and only Fr_P fixes Q(\sqrt{q^*}), i.e., if and only if Fr_P is a square in the cyclic group Gal(Q(\zeta_q)/Q). By Prop. 6.4.8 this happens exactly if p is a square modulo q.

Comparing now the 2 characterizations above about when p splits in Q(\sqrt{q^*}), we obtain quadratic reciprocity as a corollary.

04.22:  End (Prop. 6.2.1) + Discussion of differents and discriminants as measures of ramification.

I will give an application of Prop. 6.2.1, namely that we can use differents and discriminants to detect ramification. We use the presentation in this note: ramification_and_discriminants.pdf

04.20:  Prop. 4.6.6 + Prop. 6.2.1.

I overlooked the fact that we actually need Prop. 4.6.6 in the proof of Prop. 6.2.1. So we will first discuss Prop. 4.6.6.

Notes on the proof of Prop. 4.6.6: We prove this proposition in the case of completions of algebraic number fields. The separability assumption in the statement of 4.6.6 is automatic. Note 2 misprints in the proof: In the definition of \pi_m and in the representation of \gamma at the end of the argument, write f_{\alpha} at the point \alpha+\pi, and not the derivative of f_{\alpha}.

Secondly, perhaps it is difficult to see how we get from the first infinite' representation of \gamma to the finite' one at the end of the argument. See this as follows: The argument shows that for any natural number n there is \gamma_n \in o[\alpha+\pi] such that \gamma \equiv \gamma_n  mod P^n. We see then that (\gamma_n) converges to \gamma. But \gamma_n can be written in form \sum_{i= 0}^{ef-1} b_{n,i} (\alpha+\pi)^i. By Lemma 4.5.2 we deduce that - for each i - the coordinates' b_{n,i} converge in o for n converging to infinity. One deduces that the limit \gamma is in o[\alpha+\pi].

Concerning the proof of Prop. 6.2.1: We consider a Galois extension L/K of algebraic number fields. Given primes P|p of L and K we have inside Gal(L/K) the Galois group G= Gal(L_P/K_p) with its filtration G \ge G_0 \ge G_1 \ge ... . The fields that are called T and K^P in the proof of the book are, by definition the fixed fields of G_0 and G, respectively.

For the first part of the argument we use Prop. 4.6.6 and Prop. 3.12.1 to see that the different of L/T has the same valuation as the element different of \pi (a prime element for P). This can then replace the reference to Theorem 3.12.5.

After this one deduces from Theorem 3.12.3 and Theorem 4.8.5(3) that it is enough to show that the differents of T/K^P and K^P/K have valuation 0. That the different of K^P/K has valuation 0 is actually easier to see than in the book: It follows immediately from Theorem 4.8.5(3) because K^P and K have the same completion w.r.t. \nu_P.

04.15:  End (3.12 until, and including, Theorem 3.12.5 (but not the proof of 3.12.5)).

04.08 + 04.13:  Easter vacation.

04.06:  3.12 until, and including, Theorem 3.12.5 (but not the proof of 3.12.5).

I will discuss differents in the following 2 cases: In case of a finite extension of algebraic number fields, and in case of a finite extension of completions of such.

On p. 94 the module M is supposed to be a finitely generated o-module.

Concerning (3.12.6) in the proof of Theorem 3.12.4: The argument given only allows one to write \pm 1 on the right hand side of (3.12.6), but this is also enough.

You may want to refresh App. B, and perhaps also the discussion of bilinear forms and non-degeneracy of the trace form in a general setting. The notes that we used in the previous course for that are still here:  bilinear_forms.pdf

Concerning relative norms you can either read section 3.10, or these notes:  relative_norms.pdf

I will not go through the proof of 3.12.5, but just mention the result. You can read the (rather long) proof on your own if you wish. Later we will see and use a local version of this result.

04.01:  End local_global_galois_extensions.pdf.

03.30:  local_global_galois_extensions.pdf, continued.

03.25:  4.9 + Begin (6.1 + (part of 6.6)).

In 4.9 I found it easier to reverse the order of propositions 4.9.1 and 4.9.2, i.e., to show 4.9.2 first and then use it to prove 4.9.1.

The material in 6.1 and 6.6 concerns the crucial combination of Galois theory and the theory of completions that we studied in chapter 4. I want to deal with this material all at once, and I want to say more about it than what is in the book. (Also, I will employ arguments that avoid using norm maps).

I will give a development of this material as in these notes: local_global_galois_extensions.pdf  These notes can then actually replace sections 6.1 and 6.6.

03.23:  End 4.8 (Extensions of valuations on algebraic number fields).

Concerning Theorem 4.8.5: Part (1) is trivial for us since this is just the definitions of inertia degree and ramification index in the approach that we have used.

For part (3) - concerning the different' - we will simply use the definition of the different as given in the proof. The differents are certain (fractional) ideals. The is a generalized notion of a norm map over a finite field extension; the norm of the different equals then the discriminant of the extension. So part (4) follows from part (3) by applying the norm map. If the base field is Q so that this discriminant is an ideal of Z one has the theorem that is coincides with the ideal generated by the discriminant \Delta(L/Q) that we introduced in the previous course. We will later have a closer look at differents.

In the proof of Theorem 4.8.5 we have also the isomorphism (4.8.4). It is not necessary to refer to Appendix A.2 for this, - it's just the Chinese Remainder Theorem. See this note for an explanation with all details:  4.8.4.pdf  The note also explains an interesting and important consequence of the isomorphism, namely equation (3.10.2) that was not a part of the previous course (but that we now get almost for free).

03.18:  4.8 (Extensions of valuations on algebraic number fields), continued.

Concerning Prop. 4.8.4: We work with the situation: K algebraic number field, p a prime of K, and L a finite extension of K. In the proof of the proposition, the moment of truth is the claim that the ideal P defined on p. 134, line 5, is a prime ideal of O_L dividing p. That it divides, i.e., contains, p is clear. That it is a maximal ideal of O_L is seen thus: Clearly P as defined is the kernel of the map O_L --> (O_L)_{\nu} --> (O_L)_{\nu}/P_{\nu}. But according to general theory - cf. p. 109 of the book - we know that (O_L)_{\nu}/P_{\nu} is a field. Hence P is a prime ideal, - i.e., a maximal ideal (dividing p). After this one claims that (\nu_P(\alpha) = 0 implies \nu(\alpha) = 0). First, this implication is clear from definitions if \alpha is in O_L. The argument for the general case then follows on the following lines; you can - as in the book - of course refer back to Prop. 3.7.1, but it is actually just the argument that we already discussed in the note localization.pdf .

03.16:  End 4.7 (Structure of completions of algebraic number fields w.r.t. a non-archimedean valuation) + Begin 4.8 (Extensions of valuations on algebraic number fields).

Notice that the proof of Prop. 4.7.5 is actually a little simpler than in the book: By Prop. 4.7.4 we have that u^p \in U^{n+1} whenever u \in U^n. So the conclusion of the first 4 lines of the proof of Prop. 4.7.5 is clear.

After the proof of Prop. 4.7.5 I will mention (without proof) some results about the structure of the 1-unit group as a Z_p-module.

Concerning the proof of Prop. 4.7.6: We can take e_0'th roots by Prop. 4.7.5 as e_0 (being not divisible by p)  is invertible in Z_p.

03.11:  End 4.6 (Finite extensions of a complete field with a discrete valuation) + Begin 4.7 (Structure of completions of algebraic number fields w.r.t. a non-archimedean valuation).

In the beginning of the proof of Prop. 4.6.5 there is a reference to Prop. 3.10.3. At this point you should read just the proof of Prop. 3.10.3 assuming the setup of Prop. 4.6.5. Things go through verbatim, but perhaps the beginning of the proof is a bit dense; in that case you can use the following comment:  3.10.3.pdf

I will probably also mention Prop. 3.10.4 though you should read the proof of this on your own.

We skip everything in section 4.6 after the proof of Prop. 4.6.5: First, Prop. 4.6.6 is interesting but not terribly important. But you can read this remarkable statement on your own. Secondly, I do not want to go heavily into differents at this point of the course (and maybe not at all in the course); this is not because it's not important, but mainly because of time constraints. We will probably be able to do ok with merely the definition and some simple properties of differents later on.

We skip Theorems 4.7.1 and 4.7.2.

If you find the proof of Prop. 4.7.3 hard to follow you can have a look at this note:  4.7.4.pdf

03.09:  4.6 (Finite extensions of a complete field with a discrete valuation), continued.

I promised to write up the proof of Theorem 4.6.2 as presented at the lecture. However, I decided to write up a better argument which gives some more:  unramified_extensions.pdf

You can skip the remark after the proof of Lemma 4.6.3.

03.04:  4.6 (Finite extensions of a complete field with a discrete valuation), continued.

03.02:  Begin 4.6 (Finite extensions of a complete field with a discrete valuation).

I want to first continue the discussion in the note extensions_of_valuations.pdf a bit further; in particular we now see how to classify the non-archimedean valuations of algebraic number fields. Here is what I will say:  extensions_of_valuations2.pdf

After that we continue with section 4.6 as follows: At the beginning of section 4.6 the notions of relative ramification and inertia are introduced via references back to section 3.5. However, we studied section 3.5 in a slightly less general setup than what is needed now. So one option is to go back and review the slightly more general theory. That's a good idea. Nevertheless, I have instead decided to write up a little note with a more hands-on' approach to these concepts. I'm using this note in the lecture:  relative_ramification_and_inertia.pdf

02.26:  End 4.5 (Extension of valuations to finite extensions of a complete field).

Misprint on p. 122, line 6: \omega_m instead of \omega_n.

On the top half of page 123 - immediately after the proof of Theorem 4.5.3 - there is an argument for the existence of extensions of valuations (in the nonarchimedean case). In the lecture I will use a slight variant - for the cases arising from the number field situation - which probably gives a little more insight. Here is a note with the argument that I will give:  extensions_of_valuations.pdf

02.24:  End 4.4 (Hensel's lemma and refinements) + Begin 4.5 (Extension of valuations to finite extensions of a complete field).

We skip the material in section 4.4 after the proof of Prop. 4.4.6.

02.19:  4.4 (Hensel's lemma and refinements), continued.

In connection with Theorem 4.4.3 we need certain results about resultants of polynomials. I've written up a short note with the things we need here: resultants.pdf

Only 1 statement about resultants will not be proved; the proof is a simple application of the theory of symmetric polynomials + the formal properties of resultants, and you'll have to either accept it or read on your own. A possible reference is van der Waerden's book Algebra. Erster Teil.', Springer 1966. Here is a scan of the relevant pages: van_der_waerden.pdf

02.17:  I have decided to say a bit more on about the interplay between congruences and solutions of Diophantine equations in several variables. Here is a note:  diophantine_equations.pdf

02.12:  End 4.3 (Completion) + Begin 4.4 (Structure of fields complete w.r.t. discrete valuation + Hensel's lemma and refinements).

I will elaborate a bit more on Proposition 4.3.1 in the case of number fields. This will touch upon the topic of localization that we skipped in the first part of the course (section 3.7 of the book). Here is a note:  localization.pdf

In section 4.4 we will use a slightly less general setup: Instead of the arbitrary field F we will work with K_P where K is an algebraic number field and P a prime ideal of O_K. Thus K_P is the completion of K w.r.t. the exponential valuation \nu_P. I will use the following notation: We write \nu_P also for the extension of \nu_P to K_P. In K_P we have then the valuation ring \hat{O} with its maximal ideal \hat{P}. We have \hat{O}/\hat{P} \cong O_K/P which is a finite field. In this section and in the future I write O_P for \hat{O}. Also, a lot of times I will 'forget' the 'hat' in \hat{P} thus writing simply P instead of \hat{P}; it will always be clear whether P or \hat{P} is meant. You are by now sufficiently grown up mathematically to handle this abuse of notation.

I started the proof of Theorem 4.4.1 and showed that the series converges for any choice of the \alpha_i. Complete the proof of the theorem on your own as homework

Hint: Show first the following inequality: \phi( \sum_{i=i_0}^{\infty} \alpha_i \pi_i ) \le e^{-i_0} . From this one deduces that this \phi-value equals e^{-i_0} if \alpha_{i_0} is not in \hat{P} (consider the tail starting from i= i_0+1 and use the non-archimedean inequality). With this one easily completes the proof.

Notice that the proof actually gives an algorithm for determining the \alpha_i given \alpha. I will illustrate this in a couple of examples next time.

02.10:  4.2 (Valuations on the rational numbers) + Begin 4.3 (Completion).

We skip Theorem 4.2.2 about valuations on function fields.

02.05:  4.1 (Fields with valuation), continued.

Misprints: On p. 105 we clearly have n/m on the right hand side of the inequalities (4.1.1) and (4.1.2).

p. 106, line -10: Add the words for m sufficiently large'.

The argument starting on p. 106, line -6, is perhaps a bit dense. Argue as follows: Let \delta \in F. Then 1 = \phi(1) = \phi(1+\delta^m+(-\delta^m)) \le \phi(1+\delta^m) + \phi(\delta^m), so that 1-\phi(\delta)^m \le \phi(1+\delta^m) \le 1+\phi(\delta)^m. It follows that if \phi(\delta)<1 then \phi(1+\delta^m) converges to 1 for m converging to \infty. Furthermore, \phi(\delta)^m = \phi(\delta^m + 1 + (-1)) \le \phi(1+\delta^m) + 1 so that \phi(1+\delta^m) \ge \phi(\delta)^m - 1. It follows that \phi(1+\delta^m) converges to \infty if \phi(\delta) > 1. Now use these 2 conclusions with \delta = \alpha^{-1}.

02.03:  Introduction + Begin 4.1 (Fields with valuation).

You should know that what is called valuation' in the book is often called an absolute value' whereas the word valuation' denotes what is called exponential valuation' in the book. In fact, this alternative set of notation is my personal preference, but I will stick with the notation of the book. Later on, when we are very familiar with these concepts we can loosen up a bit on this point.