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Class field theory, which is so immediately compelling in its main assertions, has, ever since its invention, suffered from the fact that its proofs have required a complicated and, by comparison with the results, rather imper­ spicuous system of arguments which have tended to jump around all over the place. My earlier presentation of the theory [41] has strengthened me in the belief that a highly elaborate mechanism, such as, for example, cohomol­ ogy, might not be adequate for a number-theoretical law admitting a very direct formulation, and that the truth of such a law must be susceptible to a far more immediate insight. I was determined to write the present, new account of class field theory by the discovery that, in fact, both the local and the global reciprocity laws may be subsumed under a purely group­ theoretical principle, admitting an entirely elementary description. This de­ scription makes possible a new foundation for the entire theory. The rapid advance to the main theorems of class field theory which results from this approach has made it possible to include in this volume the most important consequences and elaborations, and further related theories, with the excep­ tion of the cohomology version which I have this time excluded. This remains a significant variant, rich in application, but its principal results should be directly obtained from the material treated here.



Chapter I. Group and Field Theoretic Foundations

Every field k comes accompanied by a canonical Galois extension: the separable algebraic closure ¯k| k. Its Galois group Gk = G(k\k) is called the absolute Galois group of k. This extension has infinite degree in almost all cases, but it has the big advantage of consolidating within it all the various finite Galois extensions of k. For this reason one would like to put ¯k| k into the forefront of Galois theoretic considerations. But one is then faced with the problem that the main theorem of Galois theory does not hold true anymore in the usual sense. We explain this by the following Example. The absolute Galois group \({G_{{\rm{I}}{{\rm{F}}_p}}} = G({\rm{I}}{{\rm{\bar F}}_p}|{\rm{I}}{{\rm{F}}_p})\) of the field IFp of p elements contains the Frobenius automorphism ϕ which is defined by
$${x^\varphi } = {x^p}\,{\rm{for}}\,{\rm{all}}\,x \in {\rm{I}}{{\rm{\bar F}}_p}$$
Jürgen Neukirch

Chapter II. General Class Field Theory

The theory which we shall develop in this chapter is of purely group theoretical nature and is concerned with an abstract profinite group G. To work in a more familiar language, however, we want to interpret G formally as a Galois group in the following sense. We denote the closed subgroups of G by G K and call the indices K fields; we refer to K as the fixed field of G K . The “field” k with G k = G is called the ground field and we denote by ¯k; the field with G¯k; = {1}. We write formally KL or L | K if GLGK and refer to the pair L|K as a field extension. L | K is a “finite extension” if G L is open (i.e. of finite index) in G K and we call
$$[L:K] = ({G_K}:{G_L})$$
the degree of the extension L|K. L|K is called normal or Galois if G L is a normal subgroup of GK. In this case we define the Galois group of L|K by
$$G(L|K) = {G_K}/{G_L}$$
Jürgen Neukirch

Chapter III. Local Class Field Theory

In this chapter we consider local fields, i.e., fields which are complete with respect to a discrete valuation and have finite residue class fields. The local fields are the p-adic number fields, i.e. the finite extensions K of the field k = Q p of p-adic numbers (case char(K) = 0), and the finite extensions K of the power series field k=F p ((x)) (case char (K) = p > 0). Here the module A K of the abstract theory will be the multiplicative group K* of K. We therefore have to study the structure of this group. We introduce the following notation. Let
  • vK be the discrete valuation of K, normalized by vK(K*) = ℤ,
  • ϑK= aεK❘vK(a)≥0 the valuation ring,
  • p K = aε K \v K (a)>0 the maximal ideal,
  • K = ϑ K /p K the residue class field, and p its characteristic,
  • U K =aεK❘v K (a) = 0 the group of units,
  • U k (n) = 1+p n K the groups of higher principal units, n=l,2,…,
  • q = q k =*k
  • ap = qVK(a) the absolute value of aεK*
  • μn the group of n-th roots of unity, and μn(K) = μ n K*. By π x , or simply π, we always mean a prime element of ϑ K , i.e. p K = πϑ K , and we set (π) = π k \kεℤ for the infinite cyclic subgroup of K* generated by π.
Jürgen Neukirch

Chapter IV. Global Class Field Theory

Let K be an algebraic number field, i.e. a finite extension of Q. A prime p of K is a class of equivalent valuations of K. We distinguish between the finite and the infinite primes. The finite primes belong to the prime ideals of K, for which we use the same notation p. The infinite primes fall into two classes, the real and the complex ones. The real primes are in 1 – 1-correspondence with the different imbeddings of K into R, and the complex primes are in 1 – 1-correspondence with the pairs of conjugate non-real imbeddings of K into C. We write p∤∞ if p is finite and p | ∞ if p is infinite, and we set S∞ = p|∞.
Jürgen Neukirch

Chapter V. Zeta Functions and L-Series

“The zeta function knows everything about the number field. We just have to prevail on it to tell us” (G. Harder). These words briefly express one of the most remarkable phenomena in number theory, namely, that many of the inner arithmetic properties of an algebraic number field are concealed in a single complex analytic function, the zeta function. The basic prototype of such a function is the Riemann zeta function
$$\zeta (s) = \sum\limits_{n = 1}^\infty {{1 \over {{n^s}}}} $$
Jürgen Neukirch


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