Complete discrete valuation rings and local fields. (cf. Chap I of [1]) Let A be a commutative ring. These are equivalent. a. A is a noetherian local ring and its maximal ideal is generated by a single non-nilpotent element. b. A is a noetherian, integrally closed, integral domain possessing a unique (≠ 0) prime ideal. c. A is a principal ideal domain possessing a unique (≠ 0) prime ideal. d. A is an integral domain; its field of fractions K possesses a “discrete valuation homomorphism” , i.e., a surjective homomorphism

v : K^{∗} → Z

such that if we make the convention that v(0), the valuation of 0, is equal to +∞ we have v(x + y) ≥ min (v(x), v(y)) for all x, y ∈ K; and moreover, A is the subset of elements of K with non-negative valuation. e. A possesses a nonnilpotent element π ∈ A such that every nonzero element a ∈ A may be written uniquely as

a = uπ^{n},

where u ∈ A^{∗} is a unit of A, and n ≥ 0. f. (Definition) A is a discrete valuation ring, DVR for short.

Some notes: The field of fractions, K, together with its valuation homomorphism, is called a discrete valued field. The quotient field k := A/m_{A} = A/πA is called the residue field of A, also “of the discrete valued field K”. Any element π ∈ K satisfying the properties of e. is called a uniformizer or a local parameter for the discrete valued field K and is a generator of m_{A}, the maximal ideal of A. The uniformizers are precisely the set of elements with valuation equal to 1. This terminology is suggested by the geometric examples of DVR’s, i.e., where A is the local ring at a smooth k-rational point x of an algebraic curve X over a field k, in which case a “local parameter” in the above sense is nothing more than a rational function on X which provides a local parameter for X about the point x.

We will deal mainly with complete discrete valuation rings. The process of completion of a DVR can be thought of either “from the perspective of A”, or “from the perspective of K”.

For the former, we consider the topology on A where a sub-base of open ideals is given by the powers of the maximal ideal (or, for that matter, we could just say: by all non-zero ideals) and form the completion

Â := proj. lim. A/m_{A}^{n},

and view Â as a topological ring. One notes that the natural ring homomorphism A → Â is an imbedding since ∩_{n=1}^{∞}m_{A}^{n} = {0}, and one proves that Â is again a DVR, and that the imbedding A → Â extends to an imbedding K → K̂ of the respective fields of fractions, which preserve discrete valuation homomorphisms.

For the latter, we just complete the discrete valued field K via a metric derived from the valuation homomorphism v (∥x∥ = exp (v(x)) will do) imbedding K in its completion K → K̂, note that the valuation v extends to a valuation v̂ on K̂, and define the DVR Â to be the elements of K̂ with nonnegative valuation, as in d. above.

“Non-archimedean” local fields of characteristic 0. We will be principally interested in complete DVR’s A = Â with field of fractions K = K̂ and residue field k, where K is of characteristic 0 and k is finite. The latter condition is equivalent to requiring that the complete DVR A is compact, or that the complete discrete valued field K is locally compact. The topology on K determines its valuation homomorphism, for the elements of A are precisely those elements which are “power-bounded” (i.e., the set of all of their powers are contained in a compact subset of K) and, for example, the maximal ideal m_{A} is the subset of K consisting of those elements whose powers converge to 0. In our discussion below we will use the term local field to refer to a complete discrete valued field K of characteristic 0 with finite residue field k (giving short shrift to the local fields R, C, and fields of positive characteristic).

How do we get local fields? First, an arbitrary finite extension L/K of a local field is again a local field (cf. Prop. 3 of section 2 of Chap I of [1]) and the topology on L is simply the topology it inherits as a finite dimensional vector space of the complete field K. Given a prime number p, any local field (of our sort) with residue field of characteristic p is a finite extension field of the field of p-adic numbers,Q_{p}, and conversely, it foollows from the above that any finite field extension K/Q_{p} is a local field. But also, you can start with a number field F/Q, consider its ring of integers O_{F} ⊂ F (i.e., the integral closure of Z in F), choose a prime ideal P ⊂ O_{F}, and form the complete DVR

O_{F,P} := proj. lim. O_{F} /P^{n},

denoting its (discrete valued) field of fractions F_{P}. Any of our local fields comes from such a construction (Exercise: show this, as an application of “Krasner’s principle”).

Galois extensions of local fields. Let L/K be a finite Galois extension of local fields with Galois group G. Denote by A ⊂ K and B ⊂ L the associated discrete valuation rings, and ℓ = B/m_{B}, and k = A/m_{A} ⊂ ℓ the respective residue fields. Let g denote the (cyclic) Galois group of the residue field extension

g := Gal (ℓ/k).

We have an exact sequence (defining the subgroup I ⊂ G):

0 → I → G → g → 0.

The Galois group G stabilizes B ⊂ L and m_{B} ⊂ B and therefore induces an action on ℓ which fixes k. This gives the homomorphism G → g. Will prove surjectivity by the elementary “Galois theoretic argument”. The subgroup I ⊂ G, the inertia group of the extension L/K, consists then of the elements of G which induce the identity automorphism on ℓ. Letting K^{ur} = L^{I} ⊂ L be the fixed field of L we have that K^{ur}/K is the maximal unramified extension of K contained in L; the residue field of K^{ur} is ℓ and Gal (K^{ur}/K) is isomorphic to g. The extension K/K^{ur} is totally ramified. We get an exact sequence:

1 → I → G → g → 1.

We get a finer analysis of G (more specifically of the inertia subgroup I ⊂ G) by considering the “lower index” filtration:

Definition. Let i ≥ 0. define the i-th ramification subgroup G_{i} ⊂ G to be the set of all elements α ∈ G such that α induces the identity automorphism on B/m_{B}^{i+1}.

Note that G_{0} = I and by convention one sometimes puts G_{−1} := G. We have defined a decreasing filtration of normal subgroups of G:

… G_{n+1} ⊂ G_{n} … G_{1} ⊂ G_{0} = I ⊂ G.

Will explain the canonical inclusion G_{0}/G_{1} ⊂ k^{∗} and the noncanonical inclusions G_{i}/G_{i+1} ⊂ k^{+} for i > 0.

Denote, provisionally, P := G_{1}, and Δ := I/P = G_{0}/G_{1} ⊂ ℓ^{∗}, the wild and tame parts of the inertia group, and note that Λ is a finite cyclic group of order dividing q − 1 where q = card (ℓ), while P is a pro-p-group, for p = the characteristic of ℓ. Put K^{tame} := L^{P}, the maximally tamely ramified subextension of L/K and denote, provisionally, T := G/P = Gal (K^{tame}/K).

We have the exact sequences:

1 → P → I → Δ → 1,

and

1 → Δ → T → g → 1.

Since Δ is a cyclic group of order prime to the order of P, there is a (noncanonical) lifting Δ → I expressing I as a semi-direct product of Δ by P.

“Functorial” construction of unramified extensions. (cf. sections 5,6 of Chap I of [1]). Let p be a prime number, and k a finite field of characteristic p, recall the Witt vector construction which associates to any perfect ring R in characteristic p, its ring W (R) of “Witt vectors”.

R ↦ W (R) := {(a_{0}, a_{1}, a_{2}, …) | a_{j} ∈ R },

with “universal p^{−∞}-polynomials” (cf. loc. cit.) defining a ring structure on W (R).

When k = F_{p}, W (F_{p}) is canonically isomorphic to the ring Z_{p} of p-adic integers, and the correspondence is simply: a = {(a_{0}, a_{1}, a_{2}, …)} ∈ W (F_{p}) ⟷ ∑_{n=0}^{∞} a_{n}p^{n} ∈ Z_{p}.

More generally, when k is a finite field, W (k) is a complete DVR with residue field canonically isomorphic to k, the reduction map being given by {(a_{0}, a_{1}, a_{2}, …)} ↦ a_{0} ∈ k. The DVR is “absolutely unramified” in the sense that p ∈ W (k) is a uniformizer. An important property of this construction, for us, is that given any complete DVR A with field of fractions K of characteristic 0, and with finite residue field k, there is a canonical ring homomorphism W (k) → A such that composition with the projection to the residue field A → k is the canonical reduction homomorphism W (k) → k described above. This ring homomorphism is an injection, and A is a free W (k)-module of rank e = the absolute ramification index of A. Moreover, we have that A may be obtained from W (k) by the adjunction of a single element π ∈ A, which is a uniformizer of A satisfying an “Eisenstein equation” over W (k), i.e., an equation of the form:

π^{e} + a_{1}π^{e−1} + … + a_{e} = 0,

where all the coefficients a_{j} are divisible by p in W (k) and the constant term a_{e} is not divisible by p^{2}.

Projective limits of the groups of units of finite fields.

First, let us review some basic facts about profinite groups which are topologically singly generated. Let Γ be such a group (which is easily seen to be abelian). Let γ ∈ Γ be a topological generator. Writing

Γ = proj. lim. Γ/N

where N ⊂ Γ runs through all open subgroups (of finite index) in Γ, one sees that for each N ⊂ Γ, the element γ projects to a generator of the group G/N, and these groups are therefore all cyclic. Moreover, any projective limit of a projective system of finite cyclic groups with surjective homomorphisms, is a group Γ such as the above. We have that Γ itself is the completion of the cyclic group < γ >⊂ Γ with respect to the topology on < γ > induced from Γ.

To get an example of the above, suppose we have a projective system of finite cyclic groups {C_{n}}_{n∈N} and such that if n a multiple of m, we have a connecting homomorphism of our projective system, C_{n} → C_{m}, and suppose that these connecting homomorphisms are all surjections. Then Γ := proj. lim. C_{n} is a group such as we have been discussing, i.e., a profinite group which is topologically singly generated. If, for some prime number p, the orders of C_{n} are all relatively prime to p, and every number relatively prime to p divides the order of some C_{n}, then an application of the Chinese Remainder Theorem gives that the corresponding group Γ is isomorphic to

∏_{ℓ≠p} Z_{ℓ},

the product being taken over all prime numbers ℓ different from p. Now let us apply this to the following situation. Let k be a finite field of cardinality q = p^{f} imbedded in an algebraic closure k̄. For each n ≥ 1, let k_{n}/k be the (cyclic, Galois) extension contained in k̄. Put w_{n} = |k_{n}^{∗}| = q^{n} − 1, and consider the projective system of finite cyclic groups C_{n} := k_{n}^{∗}, with connecting homomorphisms (when n is a multiple of m) k_{n}^{∗} → k_{m}^{∗} given by norm mappings. Since these norm mappings (i.e., , for finite field extensions) are surjective and since the orders of the C_{n}’s have the property described in the previous paragraph, if we put (provisionally) Δ_{k} := proj. lim. k_{n}^{∗}, with the projective limit compiled via the norm mappings, we have:

Δ_{k} ≈ ∏_{ℓ≠p} Z_{ℓ},

and we also have a natural continuous action of the Galois group g = Gal (k̄/k) on Δ_{k}. The reason for the symbol ≈ is that we do not have any canonical isomorphism between the two topological groups displayed above. Nevertheless, we can make up for this by noting that if k ⊂ k′ ⊂ k̄, then there is indeed a canonical identification Δ′_{k} ≅ Δ_{k} equivariant, in the evident sense, with actions of Galois groups. In particular, we have a canonical identification,

Δ_{k} ≅ Δ_{F_{p}},

for any finite field k ⊂ F̄_{p}. The g-module Δ_{F_{p}} depends, evidently, on the choice of algebraic closure F̄_{p}, but is often denoted

Δ_{F_{p}} = ∏_{ℓ≠p} Z_{ℓ}(1),

the “(1)” in the notation of the right-hand side telling us that the Galois group g as it acts on the left-hand side, while the rest of the notation reminds us of the isomorphy type of Δ_{F_{p}} as topological group. We should remember that when we adopt this notation, we have converted the group action which is written multiplicatively on the left-hand side to a group action which we write additively on the right-hand side/ For later purposes, get ready for the notation Z_{ℓ}(1) for a single given prime ℓ ≠ p, which will refer to the projection to the ℓ-adic completion of the topological g-module Δ_{F_{p}}.

“Functorial” construction of tamely ramified extensions. Fix an algebraic closure F̄_{p} of F_{p}, and for a finite subfield, k ⊂ F̄_{p} , consider the absolutely unramified extension W (k) of Z_{p} = W (F_{p}). If W (k)^{∗} denotes the (topological) group of units of the ring W (k), we have an exact sequence

0 → 1 + pW (k) → W (k)^{∗} → k^{∗} → 0.

Since the group 1+pW (k) (under multiplication) is a pro-p-group, and k^{∗} is a (finite cyclic) group of order prime to p this exact sequence splits canonically; equivalently, we have a unique homomorphism τ : k^{∗} → W (k)^{∗} lifting the projection homomorphism W (k)^{∗} → k^{∗} usually referred to as the “Teichmŭller lifting” and concretely described as follows: for ā ∈ k, choose any lifting a ∈ W (k) and define τ(ā) to be the limit of the Cauchy sequence a^{q^{n}} as n tends to ∞. it follows that if μ(W (k)) denotes the group of units in the complete DVR W (k), i.e., the torsion subgroups of W (k)^{∗}, we have μ(W (k)) ≅ k^{∗}, and in particular, W (k), and its field of fractions, K, contain a primitive q − 1-st root of unity, where q is the cardinality of k.

We now are in a position to apply the classical theory of “Kummer extensions” to construct a tamely ramified, totally ramified, extension L/K (cf. Lang’s Algebra). We will do this by extracting an w-th root of the uniformizer p. That is, we form the field extension L = K[X]/(X^{w} − p) and let π ∈ L denote the image of X. You check that L/K is indeed a Kummer extension, cyclic of degree w, with Galois group canonically isomorphic to μ(W (k)) ≅ k^{∗}. Moreover, the ring of integers in L is given by W (k)[π] and π is a uniformizer.

– Here draw the various field extensions and Galois groups involved – From this we see that Gal (L/Q_{p}) is canonically isomorphic to the semi-direct product of Gal (k/F_{p}) by k^{∗}, the action of Gal (k/F_{p}) on k^{∗} being the natural action.

Passage to the limit with tamely ramified extensions. We let K_{n} denote the field of fractions of W (F̄_{p^{n}}), K_{∞} := ∪_{n=1}^{∞}K_{n} denote the field of fractions of W (F̄_{p}). Let L_{n} denote the tamely ramified extension of K_{n} we have just constructed, and L_{∞} := ∪_{n=1}^{∞}L_{n} the union of them all.

– Here, again, draw the various field extensions and Galois groups involved – From this we see that if T := Gal (L_{∞}/Q_{p}), then we have an exact sequence,

1 → ∏_{ℓ≠p} Z_{ℓ}(1) → T → g → 1,

where the action of g on ∏_{ℓ≠p} Z_{ℓ}(1) is the natural action and, moreover, there is a continuous lifting of g to T so that T is isomorphic to the semi-direct product of g by ∏_{ℓ≠p} Z_{ℓ}(1) (with the natural action).

The structure of T : Here is a convenient “generators-and-relations” view of the profinite group T. Choose (unnaturally, of course) a topological generator τ of the group ∏_{ℓ≠p} Z_{ℓ}(1), and recall that we have our canonical generator ϕ= “Frobenius” of the group g. We have that the pair of elements {τ, ϕ} form a system of topological generators of T, and conjugation by ϕ on Δ_{F_{p}} is given by the natural action, i.e., we have the formula:

ϕτϕ^{−1} = τ^{p}.

The profinite group T may be thought of as isomorphic to the “largest” profinite group on two generators {τ, ϕ} satisfying the single relation ϕτϕ^{−1} = τ^{p}.

Exercise: Describe all continuous representations of T into GL_{2}(Z_{ℓ}) where ℓ is a prime number different from p.