TWO-REGULAR NUMBER FIELDS

AND ALGEBRAIC K-THEORY

John Rognes

March 25th 1999

In this talk I will discuss some number theory, algebraic K-theory and the Lichtenbaum–Quillen conjectures. These were proved at the prime 2 in 1996 by combining results of Suslin, Voevodsky, Bloch, Lichtenbaum, Rognes and Weibel. This gives a calculation of all higher 2-primary algebraic K-groups of rings of integers in number fields, in terms of the étale cohomology groups of the associated rings of 2-integers. In a joint paper with Østvær we make these calculations fully explicit for the class of 2-regular number fields, which include the rationals together with infinitely many quadratic and cyclotomic number fields. It is our aim to review these results today.

1. Number theory.

Let F be a number field, i.e., a finite extension of the field ℚ of rational numbers. A field homomorphism F → ℝ is called a real embedding of F, while a field homomorphism F → ℂ that does not factor through ℝ is called a complex embedding of F. The latter always occur in pairs of two distinct complex embeddings, related by complex conjugation. Let r_{1} be the number of distinct real embeddings of F, and let r_{2} be the number of complex conjugate pairs of distinct complex embeddings. Then n = r_{1} + 2r_{2} where n is the dimension of F as a rational vector space. We say that F is real if r_{1} > 0 (so F has at least one real embedding), while F is imaginary if r_{2} > 0. Also if r_{2} = 0 then F is totally real, while if r_{1} = 0 then F is totally imaginary.

Let O_{F} be the ring of algebraic integers in F. These are the roots in F of monic polynomials with integer coefficients, i.e., the solutions to equations

x^{n} + a_{1}x^{n−1} + ⋯ + a_{n−1}x + a_{n} = 0

where n ≥ 1 and all the a_{i} are integers. Algebraic number theory is largely concerned with the (always nonzero) prime ideals P ⊂ O_{F} in such number rings. When P ⊂ O_{F} is a prime ideal in O_{F}, its intersection with ℤ ⊆ O_{F} is a prime ideal in ℤ, hence has the form P ∩ ℤ = (p) for some prime number p. We say that the prime ideal P lies over the rational prime (p), and write P | (p).

Here are some examples:

(1) F = ℚ, the rational numbers themselves, have r_{1} = 1 and r_{2} = 0. The ring of integers is O_{ℚ} = ℤ, whose prime ideals are the principal ideals (p) = p · ℤ for all prime numbers p.

Typeset by AMS-TEX

(2) F = ℚ(\sqrt{d}) with d a square-free integer. When d > 0 this is a real quadratic number field with r_{1} = 2 and r_{2} = 0. When d < 0 this is an imaginary quadratic number field with r_{1} = 0 and r_{2} = 1. The ring of integers O_{F} is ℤ[\sqrt{d}] or ℤ[(1 + \sqrt{d})/2], depending on the congruence class of d mod 4. There are either one or two prime ideals in O_{F} over each rational prime (p), as determined by quadratic reciprocity. Not all of these will generally be principal.

(3) F = ℚ(ζ_{q}) with ζ_{q} = exp(2π\sqrt{−1}/q) and q ≥ 3. This is the q’th cyclotomic field. It is totally imaginary with r_{2} = φ(q)/2 where φ is the Euler phifunction. The ring of integers is ℤ[ζ_{q}], and over each rational prime (p) there may be a number of prime ideals, this time counted by the cyclotomic reciprocity law. Again these will generally not be principal.

(4) F = ℚ(ζ_{q} + ζ̄_{q}) with ζ̄_{q} the complex conjugate of ζ_{q}. This is the maximal real subfield of ℚ(ζ_{q}). It is totally real with r_{1} = φ(q)/2. The ring of integers is ℤ[ζ_{q} + ζ̄_{q}].

The number rings O_{F} are Dedekind domains, meaning that they have unique factorization of ideals. Thus every nonzero ideal 𝔞 ⊆ O_{F} admits a unique factorization as products of prime ideals

𝔞 = ∏_{P} P^{e_{P}}

where P ranges over the prime ideals in O_{F} and the exponents e_{P} are non-negative integers. The factorization is finite, i.e., only finitely many of the e_{P} are nonzero. Given a nonzero element a ∈ O_{F}, its principal ideal (a) = a · O_{F} can be factorized in this way. We write

(a) = ∏_{P} P^{e_{P} (a)}

defining the integers e_{P}(a). Clearly e_{P}(ab) = e_{P}(a) + e_{P} (b), so there are monoid homomorphisms e_{P}: O_{F} \ {0} → ℕ_{0} = ℕ ∪ {0}. They assemble to a monoid homomorphism

e: O_{F} \ {0} → ⨁_{P} ℕ_{0} .

It is generally more convenient to work with groups than with monoids. We extend the map above as follows: For a fraction a/b ∈ F with a, b ∈ O_{F} \{0} let e_{P}(a/b) = e_{P}(a) − e_{P}(b) ∈ ℤ. This defines a group homomorphism

e: F^{×} → ⨁_{P} ℤ

which fully describes the prime ideal factorization of principal ideals in O_{F}. How far is e from being an isomorphism ?

Lemma. The kernel ker(e) = O_{F}^{×} is the group of units (invertible elements) in the ring of integers O_{F}.

Definition. The ideal class group of F is the cokernel Cl(F) = cok(e).

Theorem. Let F be a number field. Its ideal class group Cl(F) is a finite abelian group.

If all prime ideals in O_{F} are principal then Cl(F) is the trivial group, so nontrivial elements in the ideal class group detect non-principal ideals in O_{F}. The order of Cl(F) is called the class number of F, denoted h_{F}.

2. ℓ-regular number fields.

It will be convenient to be able to disregard a set S of prime ideals of O_{F}, i.e., to neglect the factors in a prime ideal factorization that involve elements in the set S. Fix a prime number ℓ. We shall only be concerned with the case when S is the set of prime ideals in O_{F} that lie over the rational prime (ℓ). We define the ring R_{F} of ℓ-integers in F as the subring R_{F} = O_{F} [1/ℓ] of F.

Consider the modified group homomorphism

ē: F^{×} → ⨁_{P∤(ℓ)} ℤ

where P ranges over the prime ideals in O_{F} that do not lie over (ℓ). For these P, the P’th component of ē(a/b) is still e_{P}(a) − e_{P}(b).

Lemma. The kernel ker(ē) = R_{F}^{×} is the group of units in the ring of ℓ-integers R_{F} = O_{F} [1/ℓ].

Definition. The Picard group of R_{F} is the cokernel Pic(R_{F}) = cok(ē).

Lemma. There is an exact sequence

0 → O_{F}^{×} → R_{F}^{×} → ⨁_{P|(ℓ)} → Cl(F) → Pic(R_{F}) → 0 .

The surjection Cl(F) → Pic(R_{F}) annihilates the classes of prime ideals lying over (ℓ). In particular Pic(R_{F}) is a finite abelian group, of order dividing the class number of F.

In the case of real number fields, it will also be necessary to restrict attention to the principal ideals generated by elements in F^{×} which map to positive real numbers under all real embeddings F → ℝ.

Let F_{+}^{×} ⊆ F^{×} be the subgroup of totally positive fractions, i.e., the elements x ∈ F^{×} such that ρ(x) > 0 for each real embedding ρ: F → ℝ. When F is totally imaginary F_{+}^{×} = F^{×}, but for real F the group F_{+}^{×} is a proper subgroup of F^{×}.

Also let R_{F +}^{×} ⊆ R_{F}^{×} be the subgroup of totally positive units in the ring of ℓ-integers, given by R_{F +}^{×} = R_{F}^{×} ∩ F_{+}^{×}.

There is then a restricted group homomorphism

ē_{+} : F_{+}^{×} → ⨁_{P∤(ℓ)} ℤ .

Lemma. The kernel ker(ē_{+}) = R_{F_{+}}^{×} is the group of totally positive units in the ring of ℓ-integers R_{F} = O_{F} [1/ℓ].

Definition. The narrow Picard group of R_{F} is the cokernel Pic_{+}(R_{F}) = cok(ē_{+}).

Lemma. Let F be a number field with r_{1} real embeddings. There is an exact sequence

0 → R_{F +}^{×} → R_{F}^{×} ⟶^{σ} (ℤ/2)^{r_{1}} → Pic_{+}(R_{F}) → Pic(R_{F}) → 0 .

The signature map σ: R_{F}^{×} → (ℤ/2)^{r_{1}} takes a unit x ∈ R_{F}^{×} to its images under the r_{1} real embeddings R_{F}^{×} → ℝ^{×} followed by the homomorphism ℝ^{×} → ℝ^{×}/(ℝ^{×})^{2} ≅ ℤ/2.

Hence also the narrow Picard group Pic_{+}(R_{F}) is a finite group. For F totally imaginary it equals the Picard group, while for F real it has order at most 2^{r_{1}−1} times that of the Picard group. (Exercise: Why not 2^{r_{1}} ?)

The ℓ-rank rk_{ℓ}(A) of an abelian group A is the dimension of the kernel _{ℓ}A of the multiplication by ℓ map A → A, as a vector space over 𝔽_{ℓ}. Let t be the ℓ-rank of the Picard group Pic(R_{F}), and let u be the ℓ-rank of the narrow Picard group Pic_{+}(R_{F}). So u = t for ℓ odd, and t ≤ u < t + r_{1} for ℓ = 2. Let s be the number of prime ideals in O_{F} that lie over (ℓ).

Definition. A number field F is ℓ-regular if there is only one prime ideal in O_{F} above the rational prime (ℓ), and the narrow Picard group Pic_{+}(R_{F}) has order prime to ℓ, i.e., s = 1 and t = u = 0.

So ℓ-regular number fields have relatively simple number-theoretic behavior at the prime ℓ.

3. Quadratic and cyclotomic examples.

We now restrict attention to the case ℓ = 2 and consider 2-regular number fields. We have the following examples:

(1) F = ℚ is 2-regular, since (2) cannot split and the narrow Picard group Pic_{+}(ℤ[1/2]) is trivial.

(2) The 2-regular quadratic number fields F = ℚ(\sqrt{d}) with d assumed square free were classified by Browkin and Schinzel: F is 2-regular precisely for d = −2, −1, 2, ±p or ±2p, with p ≡ ±3 mod 8 a prime.

(3) The 2-cyclotomic number fields E = ℚ(ζ_{q}) with q = 2^{m} are all 2-regular. Here (2) is totally ramified in E, and the Picard group Pic(ℤ[ζ_{q}, 1/2]) is trivial since the class number of E is odd (by Iwasawa).

(4) For q an odd prime power, the q’th cyclotomic number field E = ℚ(ζ_{q}) is sometimes 2-regular. The rational prime (2) is inert in E precisely when 2 is a primitive root mod q, i.e., when 2 generates the group of units in ℤ/q. When additionally the Picard group Pic(ℤ[ζ_{q}, 1/2]) has odd order, E is 2-regular.

(5) The maximal real subfields F = ℚ(ζ_{q} + ζ̄_{q}) with q = 2^{m} of 2-cyclotomic number fields are all 2-regular. For the norm homomorphism Pic_{+}(R_{E}) → Pic_{+}(R_{F}) with E = ℚ(ζ_{q}) is surjective (by an argument using class field theory).

(6) For q an odd prime power such that 2 is a primitive root mod q, the maximal real subfield F = ℚ(ζ_{q} + ζ̄_{q}) of the q’th cyclotomic field E = ℚ(ζ_{q}) is 2regular when Pic(R_{E}) has odd order, or more generally when Pic_{+}(R_{F}) has odd order.

These examples account for all quadratic and cyclotomic 2-regular number fields, since (2) is always split in the q’th cyclotomic field E = ℚ(ζ_{q}) when q is not a prime power. The Picard group Pic(R_{E}) certainly has odd order when the class number h_{E} is odd, which is the case for all prime powers q with φ(q) ≤ 66, except q = 29. For more examples, recall that a Sophie Germain prime is a prime p such that also (p − 1)/2 is a prime. By a theorem of Estes, the class number of E = ℚ(ζ_{p}) is odd for Sophie Germain primes p such that 2 is a primitive root mod p. Hence the q’th cyclotomic field is 2-regular for odd prime powers q ≠ 29 such that φ(q) ≤ 66, or q = p a Sophie Germain prime, such that 2 is a primitive root mod q. Some examples are the primes q = 83, 107 and 179.

4. Étale cohomology.

Consider an extension E/F of number fields. Let 𝔭 ⊆ O_{F} be a prime ideal. Then 𝔭 · O_{E} ⊂ O_{E} is a nonzero ideal in O_{E} and admits a unique factorization in prime ideals:

𝔭 · O_{E} = ∏_{P} P^{e_{P}}

where the P range over prime ideals in O_{E}. If some e_{P} > 1 we say that 𝔭 is ramified in the extension. Otherwise, if all e_{P} ≤ 1 we say that 𝔭 is unramified. If all prime ideals 𝔭 in O_{F} are unramified, we say that the extension E/F is (finitely) unramified.

By Minkowski’s theorem, no nontrivial finite extensions of ℚ are unramified. Fix a rational prime ℓ and let R_{F} = O_{F} [1/ℓ] and R_{E} = O_{E}[1/ℓ] be the rings of

ℓ-integers. We say that E/F is unramified away from ℓ if all prime ideals 𝔭 ∤ (ℓ) in O_{F} are unramified, i.e., if all prime ideals in O_{F} that do not lie over the rational prime (ℓ) are unramified in the extension E/F. When this happens, we also say that R_{E}/R_{F} is a finite étale covering.

The ℓ-cyclotomic number fields ℚ(ζ_{ℓ^{m}}) provide examples of extensions over ℚ which are unramified away from ℓ. Thus the ℤ[ζ_{ℓ^{m}}, 1/ℓ] give examples of finite étale covers of ℤ[1/ℓ].

Consider a Galois extension E/F with Galois group G = Gal(E/F). Let μ_{ℓ^{m}}(E) ⊂ E^{×} be the subgroup of elements of order dividing ℓ^{m}, i.e., the ℓ^{m}’th roots of unity in E. Then G acts on μ_{ℓ^{m}}(E) by restriction of its action on E. The invariants of this action will be μ_{ℓ^{m}}(E)^{G} = μ_{ℓ^{m}}(F), i.e., the ℓ^{m}’th root of unity in F. We may consider the right derived functors of taking G-invariants, i.e., the group cohomology of G with coefficients in μ_{ℓ^{m}}(E):

H_{gp}^{n}(G; μ_{ℓ^{m}}(E)) := H^{n}(Hom_{G}(EG_{∗}, μ_{ℓ^{m}}(E)))

where EG_{∗} is a projective resolution of ℤ by ℤ[G]-modules, e.g, the bar resolution. More generally we can form the i-fold tensor product μ_{ℓ^{m}}(E)^{⊗i} with the diagonal G-action, and consider H_{gp}^{n}(G; μ_{ℓ^{m}}(E)^{⊗i}).

Now consider only the Galois extensions E/F that are unramified away from ℓ, and form the colimit of cohomology groups:

H_{ét}^{n}(R_{F}; μ_{ℓ^{m}}^{⊗i}) := colim_{E/F} H_{gp}^{n}(G; μ_{ℓ^{m}}(E)^{⊗i}) .

This defines the n’th étale cohomology group of R_{F} with coefficients in the étale sheaf μ_{ℓ^{m}}^{⊗i}, also written ℤ/ℓ^{m}(i).

The ℓ-adic integers ℤ_{ℓ} are the limit of the groups ℤ/ℓ^{m}. There is an extension of the definition above to étale cohomology groups with coefficients in the étale pro-sheaf {μ_{ℓ^{m}}^{⊗i}}_{m}, also written ℤ_{ℓ}(i). Thus we can talk about the ℓ-adic étale cohomology groups H_{ét}^{n}(R_{F}; ℤ_{ℓ}(i)). By a finiteness result, there are isomorphisms

H_{ét}^{n}(R_{F}; ℤ_{ℓ}(i)) ≅ lim_{m} H_{ét}^{n}(R_{F}; μ_{ℓ^{m}}^{⊗i}) .

Theorem. Let F be a number field with r_{1} real embeddings and r_{2} pairs of complex embeddings. Then

H_{ét}^{n}(R_{F}; ℤ_{ℓ}(i)) ≅ \{ \begin{matrix}ℤ_{ℓ} & for n = i = 0, \\ 0 & for n = 0, i > 0, \\ ℤ_{ℓ}^{r_{2}} ⊕ ℤ/w_{i}^{(ℓ)} & for n = 1, i ≥ 2 even, \\ ℤ_{ℓ}^{r_{1}+r_{2}} ⊕ ℤ/w_{i}^{(ℓ)} & for n = 1, i ≥ 3 odd, \\ (finite ℓ-group) & for n = 2, i ≥ 2, \\ (ℤ/(ℓ, 2))^{r_{1}} & for n ≥ 3 and i − n even, \\ 0 & for n ≥ 3 and i − n odd. \end{matrix}

The numbers w_{i}^{(ℓ)} = w_{i}^{(ℓ)}(F) are easily computed in terms of the number of ℓ-power roots of unity in F and F (\sqrt{−1}). Thus the main complexity in the étale cohomology groups of R_{F} lie in the finite ℓ-groups H_{ét}^{2}(R_{F}; ℤ_{ℓ}(i)) for i ≥ 2.

A main result in the paper of Rognes and Østvær is that for ℓ-regular number fields, H_{ét}^{2}(R_{F}; ℤ_{ℓ}(i)) is as ‘simple as possible.’ For ℓ = 2 we prove:

Proposition (Rognes–Østvær). Let F be a number field with r_{1} real embeddings. The following are equivalent:

(1) F is 2-regular.

(2) H_{ét}^{2}(R_{F}; ℤ_{2}(i)) ≅ \{ \begin{matrix}(ℤ/2)^{r_{1}} & for i ≠ 0 even, \\ 0 & for i ≠ 1 odd. \end{matrix}

(3) H_{ét}^{2}(R_{F} : ℤ_{2}(i)) ≅ (ℤ/2)^{r_{1}}.

5. Conjectures of Lichtenbaum and Quillen.

Let R be a ring, always associative with unit. To the category of finitely generated projective R-modules, Quillen associated a loop space K(R) whose homotopy groups are the algebraic K-groups of R:

K_{i}(R) = π_{i}(K(R))

for all integers i ≥ 0. These are related to algebraic number theory by the following results:

Theorem. Let F be a number field, with ring of integers O_{F} and ring of ℓ-integers R_{F}. Then K_{0}(F) ≅ ℤ, K_{1}(F) ≅ F^{×}, K_{0}(O_{F}) ≅ ℤ ⊕ Cl(F), K_{1}(O_{F}) ≅ O_{F}^{×}, K_{0}(R_{F}) ≅ ℤ ⊕ Pic(R_{F}) and K_{1}(R_{F}) ≅ R_{F}^{×}.

The arithmetic interpretation of the higher algebraic K-groups (the K_{i}(R) for i ≥ 2) is more subtle. In the 1970’s, Lichtenbaum and Quillen proposed two related conjectures on this topic. We begin with the Lichtenbaum conjecture.

The Dedekind zeta-function ζ_{F} (s) of the number fields F is first defined as a complex analytic function in the region ℜ(s) > 1 of the complex plane, by the absolutely convergent infinite series

ζ_{F} (s) = ∑_{𝔞} \frac{1}{N(𝔞)^{s}} .

Here 𝔞 ⊆ O_{F} runs over the nonzero ideals in O_{F}, and the integer N(𝔞) is the norm of 𝔞, i.e., the order of the quotient ring O_{F} /𝔞. Then one proves that ζ_{F} (s) admits a unique meromorphic extension to the entire complex plane. Lichtenbaum’s conjecture expresses the value of the extended function at negative odd integers s = 1 − 2j, for totally real abelian number fields, in terms of orders of the higher algebraic K-groups of O_{F}. (A number field F is abelian if it is Galois over ℚ, with abelian Galois group.)

Conjecture (Lichtenbaum). Let F be a totally real abelian number field with r_{1} real embeddings. Then for all even integers i ≥ 2

2^{r_{1}} · \frac{#K_{2i−2}(O_{F})}{#K_{2i−1}(O_{F})} = ζ_{F} (1 − i)

up to sign.

In particular the zeta function is takes rational values at odd negative integers. In fact, Lichtenbaum’s original conjecture did not include the factor 2^{r_{1}}, and asserted equality up to powers of 2. I have included it here because of the following theorem, which says that with this extra factor, the 2-components of the two sides agree.

Theorem (Kolster, Rognes, Weibel). Lichtenbaum’s formula is correct up to odd multiples.

This uses an extension due to Kolster of Wiles’ proof of the Main Conjecture in Iwasawa theory to relate the value of the zeta-function to a ratio of orders of two étale cohomology groups. Next it uses Rognes and Weibel’s calculation of the 2-primary algebraic K-groups in terms of étale cohomology groups, which uses Voevodsky’s proof of the Milnor conjecture on quadratic forms, and related work by Suslin, Bloch and Lichtenbaum.

Quillen’s conjecture concerned the algebraic K-groups themselves. He viewed them as arithmetic analogues of the topological K-groups of Atiyah and Hirzebruch, and conjectured that there should be a spectral sequence converging to the algebraic K-groups of a ring, similar to the Atiyah–Hirzebruch spectral sequence from the singular cohomology of a space to its topological K-theory groups. In more detail, he considered the ring R_{F} of ℓ-integers in a number field F, and replaced singular cohomology with ℓ-adic étale cohomology. Thus there should be a spectral sequence with E^{2} term

E_{s,t}^{2} = \{ \begin{matrix}H_{ét}^{−s}(R_{F}; ℤ_{ℓ}(i)) & for t = 2i \\ 0 & otherwise \end{matrix}

converging to K_{s+t}(R_{F}) ⊗ ℤ_{ℓ} for s + t ≥ 1.

6. Two-primary algebraic K-theory.

Using Suslin’s motivic cohomology and Voevodsky’s proof of the Milnor conjecture, combined with the Bloch–Lichtenbaum spectral sequence, Rognes and Weibel computed the 2-primary algebraic K-groups of the ring R_{F} of 2-integers in any number field F, in terms of the 2-adic étale cohomology groups of R_{F}.

Theorem (Rognes–Weibel). Let E be a totally imaginary number field. Then for all n ≥ 2

K_{n}(R_{E}) ⊗ ℤ_{2} ≅ \{ \begin{matrix}H_{ét}^{2}(R_{E}; ℤ_{2}(i + 1)) & for n = 2i, \\ ℤ_{2}^{r_{2}} ⊕ ℤ/w_{i}^{(2)}(E) & for n = 2i − 1. \end{matrix}

Theorem (Rognes–Weibel). Let F be a real number field. Then for all n ≥ 2

K_{n}(R_{F}) ⊗ ℤ_{2} ≅ \{ \begin{matrix}H_{ét}^{2}(R_{F}; ℤ_{2}(4k + 1)) & for n = 8k, \\ ℤ_{2}^{r_{1}+r_{2}} ⊕ ℤ/2 & for n = 8k + 1, \\ H_{ét}^{2}(R_{F}; ℤ_{2}(4k + 2)) & for n = 8k + 2, \\ ℤ_{2}^{r_{2}} ⊕ (ℤ/2)^{r_{1}−1} ⊕ ℤ/2w_{4k+2}^{(2)}(F) & for n = 8k + 3, \\ (?) & for n = 8k + 4, \\ ℤ_{2}^{r_{1}+r_{2}} & for n = 8k + 5, \\ H̃_{ét}^{2}(R_{F}; ℤ_{2}(4k + 2)) & for n = 8k + 6, \\ ℤ_{2}^{r_{2}} ⊕ ℤ/w_{4k+4}^{(2)}(F) & for n = 8k + 7. \end{matrix}

Here H̃_{ét}^{2}(R_{F}; ℤ_{2}(4k + 2)) is the kernel of the natural surjective map

H_{ét}^{2}(R_{F}; ℤ_{2}(4k + 2)) ⟶ ⨁^{r_{1}} H_{ét}^{2}(ℝ; ℤ_{2}(4k + 2)) ≅ (ℤ/2)^{r_{1}}

induced by the r_{1} real embeddings of F.

The missing group K_{8k+4}(R_{F}) ⊗ ℤ_{2} is isomorphic to the maximal finite quotient of K_{8k+5}(R_{F}; ℚ/ℤ). The latter K-group fits into the short exact sequence

0 → (ℤ/2)^{r_{1}−1} ⟶ K_{8k+5}(R_{F}; ℚ/ℤ) ⟶ (ℚ/ℤ)^{r_{1}+r_{2}} ⊕ H_{ét}^{2}(R_{F}; ℤ_{2}(4k + 3)) → 0.

For 2-regular number fields we have the following simplifications:

Theorem. Let E be a 2-regular totally imaginary number field. Then for n ≥ 2

K_{n}(R_{E}) ⊗ ℤ_{2} ≅ \{ \begin{matrix}0 & for n = 2i, \\ ℤ_{2}^{r_{2}} ⊕ ℤ/w_{i}^{(2)}(E) & for n = 2i − 1. \end{matrix}

Theorem. Let F be a 2-regular real number field. Then for n ≥ 2

K_{n}(R_{F}) ⊗ ℤ_{2} ≅ \{ \begin{matrix}0 & for n = 8k, \\ ℤ_{2}^{r_{1}+r_{2}} ⊕ ℤ/2 & for n = 8k + 1, \\ (ℤ/2)^{r_{1}} & for n = 8k + 2, \\ ℤ_{2}^{r_{2}} ⊕ (ℤ/2)^{r_{1}−1} ⊕ ℤ/2w_{4k+2}^{(2)}(F) & for n = 8k + 3, \\ 0 & for n = 8k + 4, \\ ℤ_{2}^{r_{1}+r_{2}} & for n = 8k + 5, \\ 0 & for n = 8k + 6, \\ ℤ_{2}^{r_{2}} ⊕ ℤ/w_{4k+4}^{(2)}(F) & for n = 8k + 7. \end{matrix}

This gives explicit calculations of all the 2-primary algebraic K-groups K_{i}(R_{F})⊗ ℤ_{2} and K_{i}(O_{F}) ⊗ ℤ_{2} for the examples of 2-regular number fields previously given.

From the zeta-function formula in the totally real abelian case, we can deduce the following formulas. Let v_{2}(q) = v_{2}(a) − v_{2}(b) denote the 2-adic valuation of the rational number q = a/b.

TWO-REGULAR NUMBER FIELDS AND ALGEBRAIC K-THEORY 9

Proposition. Let F = ℚ(\sqrt{d}). Then for all even i ≥ 2

v_{2}(ζ_{F} (1 − i)) = −1 − v_{2}(i)

when d = 2, while

v_{2}(ζ_{F} (1 − i)) = −v_{2}(i)

when d = p or d = 2p with p ≡ ±3 mod 8 prime. Proposition. Let F = ℚ(ζ_{2^{m}} + ζ̄_{2^{m}}) with m ≥ 2. Then for all even i ≥ 2

v_{2}(ζ_{F} (1 − i)) = 2^{m−2} − (m + v_{2}(i)).

Proposition. Let F = ℚ(ζ_{q} + ζ̄_{q}) with q an odd prime power and 2 a primitive root mod q. Suppose Pic_{+}(R_{F}) has odd order. Then for all even i ≥ 2

v_{2}(ζ_{F} (1 − i)) = φ(q)/2 − (2 + v_{2}(i)).

Department of Mathematics, University of Oslo, Norway