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and Fk* the group of units of Fk. Define ring homomorphisms ei (i = 1, , k+ 1) of Fk to Fk+l by(2'3) ei(fl? * ?fk) =fl? * fi?1 fi ... ?fk and define a multiplicative homomorphism Ak: Fk* -> F(k+l)* by Ak(X) = el(x) (E2(X)) * ... (ek+l(x))+1. Then the groups Fk* and mappings Ak form the Amitsur complex; the kth homology group Ker Ak+l/Im Ak we denote by Hk(F). In case F is a finite dimensional extension field, Amitsur showed that H2(F) is isomorphic to the Brauer group of central simple C-algebras split by F. He also showed that in case F is a normal separable extension, Hn(F) is isomorphic to Hn(G, F*) the nth cohomology group of the Galois group G of F over C with coefficients in the group of nonzero elements of F. In this paper, we extend and simplify Amitsur's results. We begin by showing (?2) that in case F is a separable field extension of C, Hn(F) --'Hn([G: H], K*), where K is a normal closure of F with Galois group G, H is the subgroup corresponding to F, and the cohomology group on the right side is the relative cohomology group as introduced in [I]. Next, we study Hn(F) when C is not necessarily a field but when n = 2. In ?3, under weak hypotheses on F, we exhibit a homomorphism of H2(F) to the (generalized) Brauer group of central separable algebra classes split by F [5]. This homomorphism is an isomorphism under stronger hypotheses on F and C. These hypotheses are slightly weaker than assuming all projective C, F, and F2 modules are free and include the cases (10) C is semilocal (not necessarily Noetherian) and F is a C-algebra which is a-finitely generated projective C-module containing C 1 as a direct summand and (20) C=K[x], F=L[x], with K a field and L a finite dimensional commutative K-algebra. Hochschild in [13] has given a description of the Brauer group in case F is a purely inseparable extension field of C of exponent 1. In [3, ?7], Amitsur

Let K K be a field having prime characteristic p p . The following conditions on a subfield k k of K K are equivalent: (i) ∩ n K p n ( k ) = k { \cap _n}{K^{{p^n}}}(k) = k and K / k K/k is separable. (ii) k k is the field of constants of an infinite higher derivation defined in K K . (iii) k k is the field of constants of a set of infinite higher derivations defined in K K . If K / k K/k is separably generated and k k is algebraically closed in K K , then k k is the field of constants of an infinite higher derivation in K K . If K / k K/k is finitely generated then k k is the field of constants of an infinite higher derivation in K K if and only if K / k K/k is regular.

Introduction. Let the difference field2 j have the transformally algebraic extension3 5. We have previously defined4 two numerical invariants, order and effective order, of such an extension. In this note we introduce a new numerical invariant, the limit degree of the extension 5 of j, which will be denoted by l.d. (5/W). We suppose, first, that 5 is finitely generated; say 5 = j(ai,* .r). Let Sk denote the set of the atj and their first k transforms. We denote by dk the degree of F(Sk+1) with respect to F(Sk), k = 0, 1, 2, * . . The dk form a nonincreasing sequence whose terms are finite integers for sufficiently large k. We define l.d. (5/j) to be the least value assumed by the dk. It will be proved in ?1 that the limit degree is independent of the choice of the set of generators ai. If 5 is any-not necessarily finitely-generated-transformally algebraic extension of j, we define l.d. (5/j) to be the maximum of the limit degrees of all finitely-generated sub-extensions of j, if this maximum exists, or oo, if it does not. That this definition is consistent with the preceding follows from Theorem I below when it is restricted to finitely generated extensions. If 9) is an irreducible manifold of dimension zero over a difference field j and 2 the corresponding prime difference ideal, we define the limit degree of 9) over j, or of 2, to be l.d. (5/ ), where 5 is the field obtained by adjoining a generic zero of 2 to j. If 5 is not transformally algebraic over F the definition of limit degree may be applied and will always result in the value oo. In this case it is more useful to consider the limit degree relative to a given basis of transformal transcendency. If S is the field formed by adjoining this basis to a, then l.d. (5/a) relative to the given basis is

Since Riemann's dissertation of 1851, Riemann surfaces have for the most part been considered as suitable domains of definition for analytic functions. Here, however, we view them as topological spaces associated with certain kinds of field extension, and consider how their topological properties are connected with algebraic properties of these field extensions. A specialization of these results gives us arithmetic properties of fields of algebraic functions of one real variable.

We give a new proof of the theorem that Amitsur’s complex for purely inseparable field extensions has vanishing homology in dimensions higher than 2. This is accomplished by computing the kernel and cokernel of the logarithmic derivativet →Dt/t mapping the multiplicative Amitsur complex to the acyclic additive one (D is a derivation of the extension field).

We define a diagram associated with a purely inseparable field extension of finite exponent. We show that, under this definition, for any given field extension the shape of its diagram is unique. Thus our diagram improves the diagram defined by Sweedler in [2, p. 402]. In §2 we define an “allowable” shape for a diagram. Given any “allowable” shape for a diagram representing a finite field extension, we construct a field extension whose diagram has that shape.

It is a classical result that if ( R , M ) (R,\mathfrak {M}) is a complete discrete valuation ring with quotient field K K , and if R / M R/\mathfrak {M} is perfect, then any finite dimensional central simple K K -algebra Σ \Sigma can be split by a field L L which is an unramified extension of K K . Here we prove that if ( R , M ) (R,\mathfrak {M}) is any regular local ring, and if Σ \Sigma contains an R R -order Λ \Lambda whose global dimension is finite and such that Λ / Rad ⁡ Λ \Lambda /\operatorname {Rad} \Lambda is central simple over R / M R/\mathfrak {M} , then the existence of an “ R R -unramified” splitting field L L for Σ \Sigma implies that Λ \Lambda is R R -separable. Using this theorem we construct an example which shows that if R R is a regular local ring of dimension greater than one, and if its characteristic is not 2, then there is a central division algebra over K K which has no R R -unramified splitting field.

In this paper, we investigate separability of CP-graded ring extensions. With restrictions neither to graded fields nor to grading by torsion–free groups, we show that some results on graded field extensions given in Hwang and Wadsworth [Commun Algebra 27(2):821–840, 1999] hold.

A proof is given for the Ruled Residue Conjecture, which asserts that if υ \upsilon is a valuation of a simple transcendental field extension K 0 ( x ) {K_0}(x) and υ 0 {\upsilon _0} is the restriction of υ \upsilon to K 0 {K_0} , then the residue field of υ \upsilon is either ruled or algebraic over the residue field of υ 0 {\upsilon _0} .

A study is made of simple central algebras over an algebraic function field of one variable with a number field of constants. It is proved that there exists an algebra with a given collection of local invariants satisfying the reciprocity law under the assumption that the orders of the invariants are odd or the field of constants is purely imaginary.

Let F be the field of algebraic functions of one variable over the field of constants k, v be a point of field F/k, and Av be the ring of functions not having poles outside point v. It is proved that Av is a GE2-ring if and only if it coincides with the ring k[X] of polynomials of one variable over field k.

In this work we construct an example of a generalized Jacobian of an elliptic curve defined over a field of algebraic numbers k such that the Serre Lie algebra p-adic representation of the Galois group of the algebraic closure of the field k in its Tate module is irreducible.

If F F is a field which is not algebraic over a finite field and G G is a polycyclic group, then all primitive ideals of the group ring F [ G ] F[G] are maximal if and only if G G is nilpotent-by-finite.

The results obtained here must have been known and settled centuries ago. However, they have proved impossible to locate in the available literature. H. K. Farahat has asked for proofs of the linear independence over the rationals of certain infinite sequences of real numbers such as √2, √3, √5.... He also raised the general question of determining the degree of the field extension generated over the rationals by a family of positive irrational numbers of the form x=a1/mwhere a, m are positive integers.

Necessary and sufficient conditions are given for two K-division rings, K an algebraic number field, to have precisely the same set of subfields. Using this, an example is presented of two K-division rings having precisely the same set of subfields such that only one of the division rings can be embedded in a Q-division ring. Let K be a field. By a K-division ring we mean a finite-dimensional division algebra with center K. If D is a K-division ring and k is a field, k'c K, we say that D is k-adequate if D can be embedded in a k-division ring. Similarly, if L is a field, we say that L is k-adequate if L is a subfield of some k-division ring. Clearly, if D is k-adequate then so is every subfield of D. In [4] the converse was raised: if every subfield of D is k-adequate, must D be k-adequate? We show that the answer to this question is no by exhibiting two K-division rings D1 and D2 having precisely the same set of subfields and such that D1 is k-adequate (and so every subfield of D2 is also k-adequate) but D2 is not k-adequate. Throughout this paper K will denote an algebraic number field. We will use freely the classification theory of K-division algebras by means of Hasse invariants. The reader is referred to [3] for the relevant theory. If a is a prime of K and D is a K-division ring, we denote the Hasse invariant of D at Y by inv,, D. The order of inv. D in Q/Z is denoted by l.i. D. Here Q denotes the field of rational numbers and Z is the ring of ordinary integers. We denote the completion of K at the prime 9 by K,. The dimension of D over K is denoted by [D: K]; we use the same notation for the dimension of field extensions. We begin by establishing criteria for two K-division rings to have precisely the same set of subfields. THEOREM 1. Let D1 and D2 be K-division rings. Then D1 and D2 have precisely the same set of subfields if and only if l.i.g Dj=l.i.,+ D2 for all primes 9 of K. Received by the editors July 19, 1971. AMS 1970 subject classifications. Primary 16A40; Secondary 12A65.

Let K be a finite Galois extension of an algebraic number field k with G = Gal (K/k), and M be a Galois extension of k containing K. We denote by resp. the genus field resp. the central class field of K with respect to M/k. By definition, the field is the composite of K and the maximal abelian extension over k contained in M. The field is the maximal Galois extension of k contained in M satisfying the condition that the Galois group over K is contained in the center of that over k. Then it is well known that Gal is isomorphic to a factor group of the Schur multiplicator H-3(G, Z), and is isomorphic to H-3(G, Z) when M is sufficiently large. In this case we call M abundant for K/k (See Heider [3, § 4] and Miyake [6, Theorem 5]).

LetKbe a field and(K) the Brauer group ofK. It consists of the similarity classes of finite central simple algebras overK. For any field extensionF/Kthere is a natural mapping(K) →(F) which is obtained by assigning to each central simple algebraA/Kthe tensor productwhich is a central simple algebra overF. The kernel of this map is the relative Brauer group(F/K), consisting of thoseA ∈(K) which are split byF.

The purpose of this note is to state the solution to the problem of local integral representation of hermitian forms when the characteristic is not 2. Hermitian and quadratic forms are two of the three main types of reflexive forms (cf. [l]). Hermitian theory of forms often follows quickly once the corresponding quadratic theory is known. However, the local integral representation theory for quadratic forms is still incomplete and finding a general solution there appears to be very difficult. Several results concerning these questions have been obtained over rings and fields of arithmetic type. Questions of fractional equivalence and representation as well as local integral equivalence have been solved for both quadratic and hermitian forms (cf. [2], [3], [4], [7]). Partial solutions to the problem of local integral representation of quadratic forms have been obtained by O'Meara [6] and Riehm [8]. Global results on the equivalence of hermitian forms have been obtained by Shimura [9]. The solution to the question of local integral representation of hermitian forms is obtained through three cases. We are given a local field E (of characteristic 7*2) with a nontrivial involution *. Let F be the fixed field of *. Then the three cases considered are those where the field extension E/F is (1) unramified, (2) ramified with E nondyadic and (3) ramified with E dyadic. The solutions in the first two cases are identical and can be stated quite easily. The third case is more complicated and some added notation will be necessary. Detailed proofs of these results will appear elsewhere.

Let k be a real field. We show that every non-negative homogeneous quadratic polynomial f (x(1),..., x(n)) with coefficients in the polynomial ring k[t] is a sum of 2n center dot tau(k) squares of linear forms, where tau(k) is the supremum of the levels of the finite non-real field extensions of k. From this result we deduce bounds for the Pythagoras numbers of affine curves over fields, and of excellent two-dimensional local henselian rings.

In this article we study in detail the category of noncommutative motives of separable algebras Sep(k) over a base field k. We start by constructing four different models of the full subcategory of commutative separable algebras CSep(k). Making use of these models, we then explain how the category Sep(k) can be described as a "fibered Z-order" over CSep(k). This viewpoint leads to several computations and structural properties of the category Sep(k). For example, we obtain a complete dictionary between directs sums of noncommutative motives of central simple algebras (=CSA) and sequences of elements in the Brauer group of k. As a first application, we establish two families of motivic relations between CSA which hold for every additive invariant (e.g. algebraic K-theory, cyclic homology, and topological Hochschild homology). As a second application, we compute the additive invariants of twisted flag varieties using solely the Brauer classes of the corresponding CSA. Along the way, we categorify the cyclic sieving phenomenon and compute the (rational) noncommutative motives of purely inseparable field extensions and of dg Azumaya algebras., Comment: 30 pages. Revised version. A strong relation between the birationality of Severi-Brauer varieties and NC motives was added

We prove that function fields of varieties of dimension at least two over an algebraic closure of a finite field are determined, modulo purely inseparable extensions, by the quotient by the second term in the lower central series of their pro-$\ell$ Galois groups., Comment: 22 pages

Let K ⊇ M ⊇ k K \supseteq M \supseteq k be a chain of fields of characteristic p ≠ 0 p \ne 0 where K K is separable over M M and M M is purely inseparable over k k . Recently it has been shown that if K K has a separating transcendency basis over M M or if M M is of bounded exponent over k k , then K = M ⊗ k S K = M{ \otimes _k}S where S S is separable over k k . This note presents an example to show that, in general, no such S S need exist.

We characterize those perfect fields k such that for each integer n > 1, but there are but finitely many isomorphism types of k-algebras of dimension n that are of finite representation type. Some remarks on the imperfect case are also presented. A finite-dimensional algebra A over a field k is of finite representation type if it has only finitely many isomorphism types of indecomposable modules. A multiplicative basis for A is a k-basis B such that B U {0} is a semigroup under the multiplication in A. Recently, Bautista, Gabriel, Roiter and Salmeron [1] have shown that an algebraically closed field k has the property that each k-algebra of finite representation type has a multiplicative basis. Let us point out at once that this property characterizes algebraically closed fields. For, if a field k has a finite extension field F, then F is of finite representation type, and any multiplicative basis B for F would be a finite semigroup with cancellation, hence a group. Then, F would be isomorphic to the group algebra kB, but kB can never be simple if B is nontrivial. An important consequence of the theorem above is that when k is algebraically closed, there are but finitely many k-algebras of finite representation type of any fixed k-dimension ("finite representation type is finite"). Let us express this by saying that k has property (N). We wish here to discuss other fields with property (N). Recall from [3, III-29] that a field k is of type (F) if it is perfect and, for each n > 1, there are only finitely many k-isomorphism types of field extensions of degree n over k. Examples are finite fields, local fields with finite residue field and the fields F((T)) of quotients of power series over algebraically closed fields F of characteristic zero. THEOREM. A perfect field k has property (N) if and only if it is of type (F). PROOF. One implication is clear. If k is of type (F), let k be an algebraic closure of k, and let G be the Galois group of k/k. By [2], a finite-dimensional k-algebra A is of finite representation type if and only if k ?k A is. Hence, it suffices to show that a k-algebra A of the form Ik ?k A has only finitely many k-forms, up to isomorphism. Such forms are classified by the set H' (G, Autkaig (A)). This set is finite by [3, III-30] since Aut-kaig (A) is a linear algebraic group defined over k, and the proof is complete. Let us remark on the case of imperfect fields. The degree of imperfection of a field k of characteristic p is that integer r so that [k: kP] = pr (or infinity, if [k: kP] is infinite). Thus, k is perfect just when its degree of imperfection is zero. It is well Received by the editors October 8, 1984. 1980 Mathematics Subject Clasification. Primary 16A46. (?)1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page

Let D ( L / K ) \mathcal {D}(L/K) denote the derivation algebra of a field extension L / K L/K of prime characteristic. If L / K L/K is purely inseparable and has an exponent, then every intermediate field F of L / K L/K equals the center of D ( L / F ) \mathcal {D}(L/F) . Here we prove the converse of this statement.

Artin's theorem on finite automorphism groups of fields extends to profinite groups, and hence every profinite group is a galois group. It is well known that every finite group is the galois group of some field extension, but the corresponding statement about profinite groups does not seem to be on record. It is proved here by generalizing Artin's theorem that finite automorphism groups are galois groups. THEOREM 1. Let F be afield and G a profinite group. Assume that (i) G acts as a group of automorphisms of F, (ii) the action is faithful, and (iii) the stabilizer of each element of F is an'open subgroup. Let K be the field offixed elements. Then F is a normal separable algebraic extension of K, and G is the galois group of the extension. PROOF. Let xl,--, xn, be any finite set of elements in F. Let Hi be the open subgroup fixing xi, so that the orbit Gxi corresponds to the H,-cosets. Let N be the intersection of all conjugates of the various Hi, a finite intersection which is then also open. The subfield L=K(Gx1," * , Gx") is mapped to itself by G, and N is the subgroup acting trivially on L; thus the finite group GIN acts faithfully on L, and the field of fixed elements is still K. By the usual Artin theorem [1, p. 194], L is a finite normal separable extension of K, and GIN is its galois group. Since F is a directed union of such fields, it is a normal separable algebraic extension of K. Thus Gal(F/K) is defined, and by assumption G maps continuously and injectively into it. Since G is compact, the image is closed; it is also dense, since G maps onto all the groups Gal(L/K). Thus G--Gal(F/K) is an isomorphism. THEOREM 2. Let G be a profinite group. Then it is the galois group of some field extension. PROOF. Let X be the disjoint union of the sets G/H for all open subgroups H. Then G acts faithfully on X, and each element of X has open Received by the editors June 7, 1973. AMS (MOS) subject classfflcations (1970). Primary 12F10; Secondary 22C05. ( American Mathematical Society 1974

Introduction. The representation theory of GL(2, K), for K a nonarchimedean local field, is somewhat more complicated than the corresponding theory for GL(2, R). In particular, the formulas for the characters of discrete series representation of GL(2, R) have a concise closed form, while the published character formulas for supercuspidal representations of GL(2, K) occupy several pages [10], [14]. The purpose of this paper is to give a concise formula for the character of certain infinite dimensional representations of GL(2, K). The representations considered are admissible in the sense that the stabilizer of a vector in the representation space is open in GL(2, K), and the subspace stabilized by an open compact subgroup of GL(2, K) is finite dimensional. Irreducible admissible representations ir of GL(2, K) satisfy Schur's lemma, so that the center K* of GL(2, K) acts by multiplication by a quasicharacter ,. There exists a locally constant character function ch , defined on a dense open subgroup of GL(2, K), which determines the isomorphism class of ir [7, Section 7]. Jacquet and Langlands [7, Section 2] introduced factors E(10 X) attached to the twist of ir by quasicharacters X of K* which describe the action of -r in a specific model. For each separable quadratic extension L of K, there exists an irreducible admissible representation BCL/K(1r) of GL(2, L), the base change of ir to L [9, Section 2]. This representation of GL(2, L) enters into the expression for the character. The main result of this paper is the following formula. Recall that conjugacy classes of nonsplit Cartan subgroups are parameterized by quadratic separable field extensions L of K, via embeddings of L* in GL(2, K).

Let L be a finitely generated nonalgebraic extensions of a field K of characteristic p # 0 and let M be a finite purely inseparable extension of L. This paper is concerned with calculating inseparability-related numerical invariants of M/K from those of L/K.

This paper deals with not necessarily maximal orders in abelian extensions of number fields. We restrict our attention to orders invariant under the Galois group G. Based on recent work of Childs and Hurley [CH], we introduce a notion of tameness for such orders (actually this is done in a slightly more general setting). The maximal order is tame in this sense if and only if the field extension is tamely ramified.

A primary problem in the theory of quadratic forms over a field F of characteristic different from two is to prove that the rings Hq*F and GWF are isomorphic. Here HqF = H*(Gal(Fq/F),Z/2Z)), where Fq is the quadratic closure of F, and GWF is the graded Witt nrng associated to the fundamental ideal of even dimensional forms in the Witt ring WF of F. In this paper, we assume we are given a field extension K of F such that WK is 'close' to WF or Hq*K is 'close' to Hq F . A method is developed to obtain information about these graded rings over F and its 2-extensions from information about the corresponding graded ring of K. This relative theory extends and includes the previously developed absolute case where K = Fq. Applications are also given to show that Hq F and GWF are isomorphic for a collection of fields anrsing naturally from the theory of abstract Witt rings.

A field is stable if for every division algebra A in its Brauer group order of A = index of A. Index and order in the BraLer group of a field F with discrete valuation and perfect residue class field K are calculated. Division algebras with specified order and index are constructed. For F complete, necessary and sufficient conditions for the stability of F are given in terms of the Brauer group of K. These results follow. A finite extension of a stable field need not be stable. The power series field K((x)) is stable for K a local field. K((x)) and K(x) are not stable for K a global field. Introduction. A field F is stable if for every division algebra A in the Brauer group Br(F), order eF(A) = index SF(A). In general order divides index and they have the same prime factors. Any local or global field is stable by class field theory. Many results concerning subfields and subalgebras of division algebras depend on stability. See, for example, [Sc] and [R]. For a profinite group G, the character group G = continuous Hom(G, Q/Z). For a field F, G(F) = Gal(FS/F) with Fs a separable algebraic closure of F. If L is a finite Galois extension of F and f is a character on Gal (L/F), then f inflates to a character on G(F). These topics are exposed in [S]. Lemma 1. Let F be a field and G = G(F). If f e G, H = Ker f, and L = fixed field of H, then L is a finite cyclic extension of F with Gal (LIF) = G/H and order f = index [ G: H] = [ L: F]. If L is a finite Galois extension of F, then L is cyclic over F iff there is a character on Gal (L/F) of order [L: F]. Proof. Immediate from Galois theory and the fact that every finite subgroup of Q/Z is cyclic. For M a field extension of F and f e G(F), fM = the restriction of f to G(M). This definition assumes Fs C Ms. If fM = ?, M is said to split f. For Received by the editors March 26, 1974 and, in revised form, May 3, 1974. AMS (MOS) subject classifications (1970). Primary 16A40, 12B20, 12G05, 12J10; Secondary 13A20, lOMIO. Copyright ? 1975, American Mathematical Society 33 This content downloaded from 157.55.39.186 on Sun, 09 Oct 2016 04:23:44 UTC All use subject to http://about.jstor.org/terms