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Local Noetherian domains arising as local rings of points of varieties or in the context of algebraic number theory are analytically unramified, meaning their completions have no nontrivial nilpotent elements. However, looking elsewhere, many sources of analytically ramified local Noetherian domains have been exhibited over the last seventy five years. We give a unified approach to a number of such examples by describing classes of DVRs which occur as the normalization of an analytically ramified local Noetherian domain, as well as some that do not occur as such a normalization. We parameterize these examples, or at least large classes of them, using the module of K\"ahler differentials of a relevant field extension., Comment: To appear in Trans. Amer. Math. Soc

We define a notion of Picard-Vessiot extension for a homogeneous linear differential equation L = 0 defined over a real differential field K with a real closed field of constants C K . When L has differential Galois group GL n over the complexification of K, we prove that a Picard-Vessiot extension for L exists over K.

Let k be an algebraically closed field, let K/k be a finitely generated field extension of transcendence degree 2 with automorphism sigma, and let A be an N-graded subalgebra of Q = K[t; sigma] with A_n finite dimensional over k for all n. Then if A is big enough in Q in an appropriate sense, we prove that GKdim A = 3,4,5 or is infinite, with the exact value depending only on the geometric properties of sigma. The proof uses techniques in the birational geometry of surfaces which are of independent interest., 26 pages, minor corrections from previous version based on referee report. To appear in Trans. Amer. Math. Soc

The Iwasawa algebra is a power series ring in one variable over the -adic integers. It has long been studied by number theorists in the context of -extensions of number fields. It also arises, however, as a ring of operations in -adic topological -theory. In this paper we study -local stable homotopy theory using the structure theory of modules over the Iwasawa algebra. In particular, for odd we classify -local spectra up to pseudo-equivalence (the analogue of pseudo-isomorphism for -modules) and give an Iwasawa-theoretic classification of the thick subcategories of the weakly dualizable spectra.

In the representation theory of finite groups, the stable equivalence of Morita type plays an important role. For general finite-dimensional algebras, this notion is still of particular interest. However, except for the class of self-injective algebras, one does not know much on the existence of such equivalences between two finite-dimensional algebras; in fact, even a non-trivial example is not known. In this paper, we provide two methods to produce new stable equivalences of Morita type from given ones. The main results are Corollary 1.2 and Theorem 1.3. Here the algebras considered are not necessarily self-injective. As a consequence of our constructions, we give an example of a stable equivalence of Morita type between two algebras of global dimension 4, such that one of them is quasi-hereditary and the other is not. This shows that stable equivalences of Morita type do not preserve the quasi-heredity of algebras. As another by-product, we construct a Morita equivalence inside each given stable equivalence of Morita type between algebras A and B. This leads not only to a general formulation of a result by Linckelmann (1996), but also to a nice correspondence of some torsion pairs in A-mod with those in B-mod if both A and B are symmetric algebras. Moreover, under the assumption of symmetric algebras we can get a new stable equivalence of Morita type. Finally, we point out that stable equivalences of Morita type are preserved under separable extensions of ground fields.

Given a connected linear algebraic group G G over an algebraically closed field C C of characteristic 0, we construct a pure Picard-Vessiot extension for G G , namely, a Picard-Vessiot extension E ⊃ F \mathcal E\supset \mathcal F , with differential Galois group G G , such that E \mathcal E and F \mathcal F are purely differentially transcendental over C C . The differential field E \mathcal E is the quotient field of a G G -stable proper differential subring R \mathcal R with the property that if F F is any differential field with field of constants C C and E ⊃ F E\supset F is a Picard-Vessiot extension with differential Galois group a connected subgroup H H of G G , then there is a differential homomorphism ϕ : R → E \phi :\mathcal R\rightarrow E such that E E is generated over F F as a differential field by ϕ ( R ) \phi (\mathcal R) .

For a prime number p p and a number field k k , let A ∞ A_\infty denote the projective limit of the p p -parts of the ideal class groups of the intermediate fields of the cyclotomic Z p \mathbb {Z}_p -extension over k k . It is conjectured that A ∞ A_\infty is finite if k k is totally real. When p p is an odd prime and k k is a real abelian field, we give a criterion for the conjecture, which is a generalization of results of Ichimura and Sumida. Furthermore, in a special case where p p divides the degree of k k , we also obtain a rather simple criterion.

We modify the abc-conjecture for number fields K in order to make the support (like the height) well-behaved under field extensions. We show further that the exponent μ > 1 of the absolute value D K of the discriminant cannot be replaced by μ = 1, and even that an arbitrarily large power of log D K must be present.

Let K K be a field and B r ( K ) Br(K) its Brauer group. If L / K L/K is a field extension, then the relative Brauer group B r ( L / K ) Br(L/K) is the kernel of the restriction map r e s L / K : B r ( K ) → B r ( L ) res_{L/K}:Br(K)\rightarrow Br(L) . A subgroup of B r ( K ) Br(K) is called an algebraic relative Brauer group if it is of the form B r ( L / K ) Br(L/K) for some algebraic extension L / K L/K . In this paper, we consider the m m -torsion subgroup B r m ( K ) Br_{m}(K) consisting of the elements of B r ( K ) Br(K) killed by m m , where m m is a positive integer, and ask whether it is an algebraic relative Brauer group. The case K = Q K=\mathbb {Q} is already interesting: the answer is yes for m m squarefree, and we do not know the answer for m m arbitrary. A counterexample is given with a two-dimensional local field K = k ( ( t ) ) K=k((t)) and m = 2 m=2 .

The notion of R-equivalence in the set X(F) of F-points of an algebraic variety X defined over a field F was introduced by Manin in [11] and studied for linear algebraic groups by Colliot-Thelene and Sansuc in [3]. For an algebraic group G defined over a field F, the subgroup RG(F) of R-trivial elements in the group G(F) of all F-points is defined as follows. An element g belongs to RG(F) if there is a rational morphism f: Al -* G over F, defined at the points 0 and 1 such that f(O) = 1 and f(1) = g. In other words, g can be connected with the identity of the group by the image of a rational curve. The subgroup RG(F) is normal in G(F) and the factor group G(F)/RG(F) = G(F)/R is called the group of R-equivalence classes. The group of R-equivalence classes is very useful while studying the rationality problem for algebraic groups, the problem to determine whether the variety of an algebraic group is rational or stably rational. We say that a group G is R-trivial if G(E)/R = 1 for any field extension E/F. If the variety of a group G is stably rational over F, then G is R-trivial. Thus, if one can establish non-triviality of the group of R-equivalence classes G(E)/R just for one field extension E/F, the group G is not stably rational over F. The group of R-equivalence classes for adjoint semisimple classical groups was computed in [15]. This computation was used to obtain examples of non-rational adjoint algebraic groups. Consider simply connected algebraic groups of classical types. Let G = GL1 (A) be the algebraic group of invertible elements of a central simple F-algebra A of dimension n2, and let H = SL1 (A) be the subgroup in G of the reduced norm 1 elements. The group H is a simply connected group of inner type A,-1. V. VoskresenskiY [29] has shown that the natural homomorphism Ax -* K1 (A) induces an isomorphism

We give homological conditions which determine sectional category, secat, for rational spherical fibrations. In the odd dimensional case the secat is the least power of the Euler class which is trivial. In the even dimensional case secat is one when a certain homology class in twice the dimension of the sphere is −1 times a square. Otherwise secat is two. We apply our results to construct a fibration p such that secat(p) = 2 and genus(p) = ∞. We also observe that secat, unlike cat, can decrease in a field extension of Q.

This paper presents an indecomposable finite-dimensional division algebra of p-power index that decomposes over a prime-to-p degree field extension, obtained by adjoining p-th roots of unity to the base. This shows that the theory of decomposability has an arithmetic aspect. Suppose F is a field and D is an indecomposable F-division algebra, that is, a division algebra that cannot be expressed as the tensor product of two nontrivial F-division algebras. It is easy to see that the (Schur) index of D must be a power of some prime p. In "Problem 6" of [Sa], Saltman asks if in general D remains indecomposable upon arbitrary prime-to-p extension. At issue is the nature of indecomposability, in particular whether or not it is "geometric". For example in [K], Karpenko showed a certain generic class of division algebras are indecomposable by computing the degrees of cycles on their Brauer-Severi varieties. As noted in [K], it is immediate from the geometric nature of the proof that these algebras remain indecomposable over all prime-to-p extensions. This paper presents an indecomposable division algebra that decomposes over a prime-to-p extension, namely the cyclotomic extension defined by p-th roots of unity. Thus it is proved that (in)decomposability can have an arithmetic aspect. Let p be an odd prime of Q, let k be a number field that does not contain a pth root of unity, and let k[s, t] be the polynomial ring in two variables over k. Define v :k[s, t] -+Z fDlZ, f --* (a, b) where b is smallest such that f E (tb) and a is smallest such that f E (Sa,tb+l). The map v is a valuation, with value group Z 20 Z ordered reverse lexicographically, so (a, b) < (a', b') if b < b', or if b = b' and a < a'. The field of iterated power series F =k((s))((t)) is Henselian with respect to v, with valuation ring R = k[[s]] +tk((s))[[t]] c k((s))[[t]] . R is a non-Noetherian 2-dimensional Henselian local ring, with maximal ideal (s) = (s, t) and residue field k. The ideal (s) properly contains the (infinitely generated) prime ideal tk((s))[[t]] = (t, t, I,. t Received by the editors June 10, 1998 and, in revised form, October 6, 1998. 1991 Mathematics Subject Classification. Primary 16K20; Secondary 1R37. ?)2000 American Mathematical Society

Let q be a prime power and let Fq be the finite field with q elements. For each polynomial Q(T) in lFq[T], one could use the Carlitz mo(lule to construct an abelian extension of 1Fq(T), called a Carlitz cyclotomic extension. Carlitz cyclotomic extensions play a fundamental role in the study of abelian extensions of 1Fq(T), similar to the role played by cyclotomic numrLber fields for abelian extensions of Q. We are interested in the tower of Carlitz cyclotomic extensions corresponding to the powers of a fixed irreducible polynomial in 1Fq[T]. Two types of properties are obtained for the 1-parts of the class numbers of the fields in this tower, for a fixed prime number 1. One gives congruence relations between the I-parts of these class numbers. The otsher gives lower bound for the 1-parts of these class numbers. Systematic study of cyclotomic field extensions of rational numbers started in the nineteenth century with Kummer and was essential in his work on Fermat's Last Theorem. Towers of cyclotomic number fields were first investigated by Iwasawa in the mid 1950's. One major application of his theory is to determine the growth of the p-divisibility of the class numbers for the fields in the tower [1w]. The study of the cyclotomic theory of function fields started with Carlitz [Ca] in 1930. Let p be a prime and let q be a power of p. Carlitz regarded the rational function field k IFq (T) and the associated polynomial ring A -Fq [T] as analogs of the rational number field Q and its ring of integers Z. He constructed an Amodule, later called the "Carlitz module" , out of the completion of the algebraic closure of IFq((T)), an analog of the field of complex numbers. For each polynomial P in A, one could use the Carlitz module to construct a field extension k(P) of k. The extensions obtained this way are called cyclotomic extensions and are essential in the study of all abelian extensions of k. Fix an irreducible polynomial P in A, and let n run through the set of positive integers. The cyclotomic extensions of k associated to Pn via the Carlitz module form a tower of extensions: nk c k(P) C k(p2) C .. C k(Pn) C It would be interesting to study the growth of the p-divisibility of the the class numbers for these fields along this tower, as Iwasawa did for cyclotomic extensions of a number field. This is the problem we would like to investigate in this paper. As in the case of a cyclotomic number field, one can decompose the class number h(k(Pn)) of k(Pn) into two integer factors h+(k(Pn)) and h-(k(PT)), called the real part and the relative part of the class number. Let p be the unique prime Received by the editors May 15, 1997. 1991 Mathematics Subject Classification. Primary 1IR29, 11R58; Secondary 11R23.

Let F be a finite field, A = F[T], and k = F(T). Let Km = k(Am) be the field extension of k obtained by adjoining the m-torsion on the Carlitz module. The class number hm of Km can be written as a product hm = h+ hThe number hm is called the relative class number. In this paper a formula for hm is derived which is the analogue of the Maillet determinant formula for the relative class number of the cyclotomic field of p-th roots of unity. Some consequences of this formula are also derived. Let Q denote the rational numbers and consider the cyclotomic field Kp = Q((p) with class number hp. It is well known that this class numbers factors as a product h+hp of two integers. The number h+ is the class number of the maximal real subfield of Kp. The integer hp is called the relative class number. In [Ca] and [Ca-O] it is shown how the relative class number can be computed in terms of a certain classical determinant known as the Maillet determinant. A nice exposition of this is given in Chapter 3 of [L] (see Theorem 7.1). In this note we will give an analogue of this material in the context of cyclotomic function fields. Let F be a finite field with q elements and A = F[T] the polynomial ring over F. Let k = F(T) be the quotient field of A. For an irreducible polynomial m of degree d we will denote by Am the m-torsion on the Carlitz module and let Km = k(Am) be the "cyclotomic" function field obtained by adjoining the elements of Am to k. For the definition of the Carlitz module and its properties see [H] and [G-R]. The class number of Km, hm, factors as a product of two integers hm = h+ h-, where h+ is the class number of the "maximal real" subfield of Km, i.e. the decomposition field of the prime at infinity of k in Km, and his called the relative class number. Our aim is to give an expression for has a product of certain easily computed determinants related to the classical Maillet determinant. We begin by recalling the analytic class number formula for hwhich follows immediately from Theorem 2 of [G-R]. A character X of (A/mA)* is said to be real if its restriction to F* is the trivial character. Otherwise it is said to be non-real or imaginary. If X is imaginary, define S(W) = Za X(a), where the sum is over all monic polynomials of degree less than d. Then, (1) ]Jm = I S(W) X imaginary Received by the editors July 2, 1995 and, in revised form, November 15, 1995. 1991 Mathematics Subject Classification. Primary 11R29; Secondary 11R58, 14H05. This work was partially supported with a grant from the National Science Foundation. (?)1997 American Mathematical Society 1299 This content downloaded from 157.55.39.215 on Tue, 30 Aug 2016 04:47:14 UTC All use subject to http://about.jstor.org/terms

Let A A be a ring with a finite group of automorphisms G G , and let f 1 {f_1} and f 2 {f_2} be homomorphisms from A A into some division algebra D D such that f 1 {f_1} and f 2 {f_2} agree on the fixed ring A G {A^G} . Assuming certain additional assumptions, it is shown that f 1 {f_1} and f 2 {f_2} differ only by an automorphism in G G and an inner automorphism of D D .

We examine the action of the general linear group GL L ( V ) {\text {GL}}_L(V) on the set of all K K -subspaces of V V , where L / K L/K is a finite field extension and V V is a finite-dimensional vector space over L L . The orbits are completely classified in the case of quadratic and cubic extensions; for infinite fields, the number of orbits is shown to be infinite if the degree of the extension is at least four. As an application we obtain q q -analogues of tranformation and evaluation formulas for hypergeometric functions due to Gessel and Stanton.

We present a short proof of a theorem by Serre on the trace form of a finite separable field extension. Let M/K be a finite Galois extension in characteristic 7 2, and assume that M is the splitting field over K of an irreducible polynomial f(X) E K[X] of degree n. We embed the Galois group G = Gal(M/K) transitively into Sn by considering the elements of G as permutations of the roots of f(X). From the 'positive' double cover 1 -) ,L2 S> Sn 1 of Sn (i.e., the double cover in which transpositions lift to elements of order 2) we then get an extension (*) 1 L G+ G -2 1 of G with the cyclic group L2 = {?1}. Let y+ E H2(G, u2) be the characteristic class of (*). We embed Sn into the orthogonal group On(Ksep) as permutations of the standard basis vectors el,..., en E Ksp. (Ksep being the separable closure of K.) As the pre-image in the Clifford group Cn(Ksep) of a transposition (i j), i < j, we can then take the element xij = (e ej)I/x/. The subgroup of Cg(Ksep) generated by these is exactly the double cover S+ of Sn, and we get a diagram Gal(K)

Let K be an uncountable field, let K C F be a field extension, and let A be an associative K-algebra. We show that if F OK A contains a non-commutative free algebra, then so does A. Throughout this note K will be a field and Ko will be the prime subfield of K. Let A be an associative K-algebra. By this we mean, in particular, that K is contained in the center of A and IK = 1A We would like to know whether or not A contains (a) a free semi-group and (b) a free K0-algebra. Both of these free objects are presumed to be non-commutative on two generators. For a more detailed discussion of free subobjects of associative algebras we refer the reader to [LI], [L2], [LM] and [K]. The following result was conjectured by Makar-Limanov. Conjecture 1. Let D be a skew field, let K be a subfield of its center, and let F be a field extension of K. If F OK D contains a free Ko-algebra, then so does D. In this note we prove this conjecture under the additional assumption that K is an uncountable field. Our main result is the following theorem. Theorem 1. Let K be an uncountable field, A an associative K-algebra, and F a field extension of K. Denote the common prime field of K and F by Ko. (a) If F OK A contains a copy of the free semi-group, then so does A. (b) If F OK A contains a copy of the free Ko-algebra, then so does A. We remark that by [LM, Lemma 1] elements x, y E A generate a free subalgebra over Ko if and only if they generate a free subalgebra over K. Note that since we are assuming IA = IK, the argument of [LM, Lemma 1] goes through even if A is not a domain. Our proof of Theorem 1 is based on a general position argument. The condition that a pair of elements generates a free object in A is given by a countable number of polynomial inequalities; see Lemmas 2 and 3. Thus over an uncountable field the set of all such pairs behaves very much like an open set in the Zariski topology. In particular, we can prove the existence of a K-point by exhibiting an F-point. We now make these ideas precise. Lemma 1. Let K be an uncountable field and let X1, X2, ... be a countable number of Zariski closed subsets of K'. If U,1 Xi = K'm then Xi = K' for some i > 1. Received by the editors April 11, 1994 and, in revised form, June 24, 1994. 1991 Mathematics Subject Classification. Primary 16S10; Secondary 20M05. (?)1996 American Mathematical Society

As is well known, Nq (n) = (1/n) Edin g(d)qn/d coincides with the number of monic irreducible polynomials of GF(q)[X] of degree n . In this note we discuss the curve nNx(n) and the solutions of equations nNx(n) = b (b > n) . As a corollary of these results, we present a necessary and sufficient arithmetical condition for R/K to have a primitive element. 0. INTRODUCTION Throughout this paper, K means a finite field, and all ring extensions of K are assumed to be commutative and have an identity that is contained in K. Moreover, all Galois extensions mean that in the sense of [ 1]. A Galois extension R/K is called simple if R is K-algebra isomorphic to a factor ring K[X]/(h) for some polynomial h in K[X], that is, R/K has a primitive element. In [4, 6, 7] and etc., the authors made some studies on primitive elements of Galois extensions from several angles. On the other hand, the simplicity of separable extensions was recently discussed by J. -D. Therond [14] in some directions. But, conditions studied in [14] are necessary and sufficient conditions so that "all" separable extensions of a semilocal ring have primitive elements. Hence, these conditions are not always applicable to discuss whether a given Galois extension is simple or not. The purpose of this note is to study the solutions of a certain equation, which is concerned with finite fields and, using these results, to present arithmetical conditions for the simplicity of Galois extensions over K. In ? 1 we consider a polynomial of degree m: Nx(m) = (1 /m) Z Iu(d)Xm/d dim where ju is the Moebius function on the set of natural numbers. As in [9], for a finite field GF(q) with q a power of a prime number, Nq(m) is the number of monic irreducible polynomials of GF(q)[X] of degree m. The aim Received by the editors May 30, 1990 and, in revised form, December 12, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 13B05; Secondary 13B25, 12E1 2. ? 1992 American Mathematical Society 0002-9939/92 $1.00 + $.25 per page

Let L be a field of characteristic p ≠ 0 p \ne 0 . A subfield K of L is Galois if K is the field of constants of a group of pencils of higher derivations on L. Let F ⊃ K F \supset K be Galois subfields of L. Then the group of L over F is a normal subgroup of the group of L over K if and only if F = K ( L p r ) F = K({L^{{p^r}}}) for some nonnegative integer r. If L / K L/K splits as the tensor product of a purely inseparable extension and a separable extension, then the algebraic closure of K in L, K ¯ \bar K , is also Galois in L. Given K, for every Galois extension L of K, K ¯ \bar K is also Galois in L if and only if [ K : K p ] > ∞ [K:{K^p}] > \infty .

Let R R be a ring with identity and M M a two-sided R R -module. It is shown that every right quotient ring in the sense of Gabriel of the trivial extension of R R by M M is a trivial extension of a right quotient ring of R R by a suitable two-sided module in case R M _RM is flat and finitely generated by elements which centralize with every element of R R .

The notion of defect for finite algebraic extensions of valued fields is classical and due to Ostrowski. Recently Matignon has generalized Ostrowski's definition to rk 1 (residually transcendental) valued function fields and used it to prove a very sharp version of the genus reduction inequality for 1dim function fields. The further generalization of the notion of defect to valued function fields of arbitrary rk is treated here. Let (K/Ko , v) be a valued function field of dim n, i.e., K/Ko is a finitely generated field extension of deg of transcendence n and v is a valuation of K. Let V5 c V, ko c k, and Go c G be the respective valuation rings, residue fields, and value groups of the extension Ko c K; and let * denote image under the v-residue map V Vl/mr = k. A transcendence basis t = {t1, ... , tn} of K/Ko will be called a residually transcendental (abbreviated tr.) basis of the valued function field if v(ti) > 0 (i = 1, ...,n) and the set of v-residues t* = {t*, ...,t} is algebraically independent over ko. The function field will be called residually tr. if there exists a residually tr. basis. A transcendence basis t is residually tr. iff v IKo(t) is the inf extension, denoted vt, of v0 w.r.t. t, i.e., iff for all f(t) in KO[t], v(f) = the inf of the values of the coefficients of f. Note that the value group of vt is clearly GO) and the residue field is ko(t*). (Cf. [3, p. 161, Proposition 2].) If t is a residually tr. basis, the henselian defect at t is defined to be D (t) := [K h: KO(t)h]/IR, where Kh denotes henselization, I = [G: GO], and R = [k: ko(t*)]. We shall prove here the 0. Independence Theorem. Let (K/KO , v) be a residually tr. valued function field. Then Dh (t) is independent of the choice of residually tr. basis t. The case that K/Ko is simple tr. has been proved in [13, Theorem 2.2]. Also, Matignon [10, p. 191, Corollary 1] has proved that if rk v = 1, then the Received by the editors July 5, 1988. 1 980AMathematics Subject Classification (1985 Revision). Primary 1 3A18, 12F20.

The goal of this paper is to introduce some structural ideas into the hitherto chaotic subject of infinite inseparable field extensions. The basic discovery is that the theory is closely related to the well-developed study of primary abelian groups. This analogy undoubtedly has implications beyond those included here. We consider only modular extensions, which are the inseparable equivalent of galois extensions. §§2 and 3 develop the theory of pure independence, basic subfields, and tensor products of simple extensions. The following sections are devoted to Ulm invariants and their computation; the existence of nonzero invariants of arbitrary index is proved by means of a theorem which furnishes an actual connection between primary groups and inseparable fields. The final section displays some complications in the field extensions not occurring in abelian groups.

Let S be a commutative, separable algebra over the commutative ring R and finitely generated and projective as an R-module. Suppose G is a group of ring automorphisms of S stabilizing R setwise. It is shown that for the ring of invariants S G {S^G} to be a strongly separable extension of R G {R^G} it is necessary that R ⋅ S G R \cdot {S^G} be R-separable; and it is shown that this condition is sufficient when R and S are finitely generated algebras over an algebraically closed field and G is a linearly reductive algebraic group acting rationally on S.

A proof of the following theorem is given: If M \mathcal {M} is a finitely generated extension of a partial difference field K \mathcal {K} then every subextension of M / K \mathcal {M}/\mathcal {K} is finitely generated. An integral measure of partial difference field extensions having properties similar to the dimension of field extensions and the limit degree of ordinary difference field extensions and a new method of computing transformal transcendence degree are developed.

Any finite degree field extension K / F K/F determines an ideal T K / F {\mathcal {T}_{K/F}} of the Witt ring W F WF of F F , called the transfer ideal, which is the image of any nonzero transfer map W K → W F WK \to WF . The ideal T K / F {\mathcal {T}_{K/F}} is computed for certain field extensions, concentrating on the case where K K has the form F ( a 1 , … , a n ) F\left ({\sqrt {{a_1}} , \ldots ,\sqrt {{a_n}} } \right ) , a i ∈ F {a_i} \in F . When F F and K K are global fields, we investigate whether there is a local global principle for membership in T K / F {\mathcal {T}_{K/F}} . This is shown to be equivalent to the existence of a "Hasse norm theorem mod squares," i.e., a local global principle for the image of the norm map N K / F : K ∗ / K ∗ 2 → F ∗ / F ∗ 2 {N_{K/F}}: {K^\ast }/{K^{\ast 2}} \to {F^\ast }/{F^{\ast 2}} . It is shown that such a Hasse norm theorem holds whenever K = F ( a 1 , … , a n ) K = F(\sqrt {a_1},\ldots ,\sqrt {a_n}) , although it does not always hold for more general extensions of global fields, even some Galois extensions with group Z / 2 Z × Z / 4 Z \mathbb {Z}/2\mathbb {Z} \times \mathbb {Z}/4\mathbb {Z} .

Let G = F ⟨ η 1 , … , η n ⟩ \mathcal {G} = \mathcal {F}\langle {\eta _1}, \ldots ,{\eta _n}\rangle be a finitely generated extension of a differential field F \mathcal {F} with m derivative operators. Let d be the differential dimension of G \mathcal {G} over F \mathcal {F} . We show that the numerical polynomial \[ ω η / F ( X ) − d ( X + m m ) {\omega _{\eta /\mathcal {F}}}(X) - d\left ( {\begin {array}{*{20}{c}} {X + m} \\ m \\ \end {array} } \right ) \] can be viewed as the differential dimension polynomial of certain extensions. We then give necessary and sufficient conditions for this numerical polynomial to be zero. An invariant (minimal) differential dimension polynomial for the extension G \mathcal {G} over F \mathcal {F} is defined and extensions for which this invariant polynomial is d ( X + M m ) d\left ( {\begin {array}{*{20}{c}} {X + M} \\ m \\ \end {array} } \right ) are characterised.

Let A be a central separable algebra over a commutative ring R. A proper R-subalgebra of A is said to be maximal if it is maximal with respect to inclusion. Theorem. Any proper subalgebra of A is contained in a maximal one. Any maximal subalgebra B of A contains a maximal ideal m A \mathfrak {m}A of A, m \mathfrak {m} a maximal ideal of R, and B / m A B/\mathfrak {m}A is a maximal subalgebra of the central simple R / m R/\mathfrak {m} algebra A / m A A/\mathfrak {m}A . More intrinsic characterizations are obtained when R is a Dedekind domain.

Let K ⊃ k K \supset k be fields having characteristic p ≠ 0 p \ne 0 . The following is proved. If K K is algebraic over k k then K K is modular over k k if and only if K = S ⊗ k M K = S{ \otimes _k}M where S S is separably algebraic over k k and M M is purely inseparable, modular. If K K is finitely generated over k k (not necessarily algebraic), then K K is modular over k k if and only if K K where M M is finite, purely inseparable, modular over k k , and S S is a finitely generated, separable, extension of k k . This leads immediately to the representation K = ( S ⊗ k M ) ⊗ S R K = (S{ \otimes _k}M){ \otimes _S}R where S S is finite separable over k , M k,\;M is finite, purely inseparable, modular over k k and R R is a regular finitely generated extension of S S , This last representation displays subfields of K / k K/k related to recently obtained Galois theories. The above results are used to analyze transitivity properties of modularity.

A formally real field of algebraic numbers is constructed which has decidable elementary theory and does not have a real closed or p-adically closed subfield. Introduction. In his list of problems [7], A. Robinson remarked (p. 501, loc. cit.): "I do not know of any proper subfield of the field of algebraic numbers, other than the fields of algebraic real orp-adic numbers, that has been shown to be decidable". Taken literally, this remark is rather strange, because the well-known results of Ax-Kochen-Ersov of 1964-1965 provide several decidable fields of algebraic numbers other than the fields mentioned by Robinson. But each of these is henselian with respect to a certain nontrivial valuation, so has a p-adically closed subfield for some prime p. (See [3] for the notion of p-adically closed field. A field of algebraic numbers is p-adically closed iff it is isomorphic with the field of algebraic p-adic numbers, similarly as a field of algebraic numbers is real closed iff it is isomorphic with the field of real algebraic numbers.) It is also easy to see that a field extension of finite degree over a decidable field of algebraic numbers is a decidable field. But applying this result to one of the fields indicated above gives again fields with ap-adically closed or real closed subfield. So probably Robinson wanted a decidable field of algebraic numbers which has no p-adically closed or real closed subfield. In ?2 we will construct such fields. I am indebted to Jan Denef for calling my attention to the question answered in this paper. 1. Preliminaries. In this and the next section, n is a fixed integer larger than 1. We define OF,, as the 1st order theory whose models are the structures (K, P1, . .. , P,) with (K, Pi) an ordered field, i.e. K is a field and P, + P, c PiPi*P, c P,, P, n P, = {0}, P, U (-Pi) = K(I < i

This article examines the concept of “analytic independence". Several illustrative examples have been included. The main results are Theorems 1-4 which state the relations between analytic independence and the degree of field extensions, transcendence degree, order of poles and “gap” respectively.

Suppose that C is an algebraically closed field and that Q is a subfield of C. If S is a nonempty subset of C disjoint from Q, it follows from an application of Zorn's lemma that there is a subfield k of C which is maximal with respect to the properties that QCk and k and S are disjoint. The problem is to describe the field extension C/k. When S consists of a single element this has been done by Quigley [4, Theorems 1, 2 and 3]. In this note we shall give several theorems which describe C/k when S consists of exactly two elements. When S contains more than two elements, some of the arguments used in the proof of Theorem 2 fail. The first theorem holds when S is any finite (nonempty) subset of C disjoint from Q. I t generalizes one of Quigley's results [4, Lemma 1].

Introduction. Let C be a field of characteristic p 0, and let K be a finite algebraic extension field of C such that the pth power of every element of K lies in C. Then K/C is called a purely inseparable extension of exponent 1. It was shown by N. Jacobson that there is a Galois theory for such extensions in which the place of the Galois group is taken by the derivation algebra of K/C. In particular, if K is any field of characteristic p, the purely inseparable extensions K/C of exponent 1 are precisely those in which C is the field of constants of a restricted K-Lie ring of derivations of K which is of finite dimension over K. In the classical theory of simple algebras, it is shown that, if K/C is a Galois extension, the Brauer similarity classes of the simple algebras with center C and split by K constitute a group which is canonically isomorphic with the group of equivalence classes of the group extensions of the multiplicative group of K by the Galois group of K/C. The present paper provides the answer to a question put to me by J-P. Serre, of whether one could establish an analogous result, for K/ C purely inseparable of exponent 1, in which restricted Lie algebra extensions [2] of K by the derivation algebra of K/C take the place of the group extensions. Not only is the answer to this question affirmative, but it provides an excellent illustration of the theory of restricted Lie algebra extensions. It turns out, in fact, that the Lie algebra extensions which arise from simple algebras are trivial extensions when regarded as ordinary extensions, so that the essential structural elements are here precisely those which differentiate the restricted extensions from the ordinary ones. ?1 contains the field theoretical background of our problem. In particular, it gives a simple proof of the main theorem of Jacobson's Galois theory [4] which we include here because it gives us the connection, on which many of our subsequent arguments are based, between the structure of the field extension K/C and that of the derivation algebra of K/C. Theorem 2, which is not needed in the sequel, is the analogue for the present situation of a well known result in the classical Galois theory and is significant for the cohomology theory of derivation algebras. In ?2 we give a proof of a theorem of Jacobson's on derivations (in a slightly generalized form) which is fundamental for the crossed product theory that follows, in the same way as the analogous theorem for isomorphisms is the source of the classical theory of crossed products. In ?3 we discuss the special type of restricted Lie algebra

It is well known that if F is a field of characteristic zero and K= F(cq, ..., ,Cc) is a finite algebraic extension of F, then K contains a primitive element, i.e. an element a such that F(al, . . ., a() = F(x). Moreover, by means of Galois theory, it is possible to characterize those elements of the extension field which are primitive. In the case of finite differentially algebraic extensions the theorem without further restrictions is false. Let Q be the field of rational numbers and 8 the usual derivation, i.e., 8q=O for every q E Q. Let c1, . . ., cn, be algebraically independent complex numbers over Q. If (Q , 8) is the differentially algebraic extension of Q where 8c=0 for every c E Q , then the underlying set of Q is identical with that of Q(c1, . . ., cs), whence it is clear that there is no element c E Q such that Q = Q . Kolchin [2] (also [5, p. 52]) has shown the existence of primitive elements in the case where the differential field F has one derivation operator and the field F has an element f such that 8f# 0(1). The differential fields (F , D) considered in this paper are differentially algebraic over F, but F does not contain nonconstant elements. We prove the existence of primitive elements in the case where the derivation operator satisfies the conditions

Classical Galois cohomological results for purely inseparable field extensions of exponent one are generalized here to commutative rings of prime characteristic. Given a commutative ring extension C over A of prime characteristic p, there are three variants for the Brauer group B(C/A) of central separable A-algebras split by C: the Amitsur cohomology group H2(C/A, Gm), the Chase-Rosenberg group PV(C/A), and Hochschild's group 4(C, g) of regular restricted Lie algebra extensions of C by the Lie algebra g of all A-derivations on C. In this paper we show that if C is finitely generated projective as an A-module and C [g] = EndA (C), then H2(C/A, Gm) , C) ,l(C/A). As a corollary we show that Hi(C/A, Gm) is zero for all i > 2. When C is a field, these are the results of Berkson, Hochschild and Rosenberg and Zelinsky [4], [11], [12]. As in [11] we show that the Lie algebra extensions which arise from central separable algebras are trivial extensions when regarded as ordinary extensions so that the essential structural elements are here precisely those which differentiate the restricted extensions from the ordinary ones. We also show that if R is a commutative C-algebra which is finitely generated, projective as a C-module, then the Brauer group B(R/A) is mapped onto the Brauer group B(R/C). The last result is also due to Hochschild when C is a field [10]. ?1 contains the background on projective modules which came into the picture. Due to their peculiar behavior all relevant automorphisms turn out to be inner which explains why instead of some exact sequences we get two isomorphism theorems. In ?2 the isomorphism of e(g, C) with 91(C/A) is proved. ?3 and ?4 provide the preliminary materials for ?5. ?3 contains an exposition on the theory of differentials in rings of prime characteristic. Its application to Amitsur cohomology is given in ?4. The main results are given in ?5. Throughout this paper C over A always denotes a commutative ring extension of prime characteristic p such that C is finitely generated projective as an A-module Received by the editors October 30, 1969. AMS 1970 subject classifications. Primary 13A20.

An inversive difference field K \mathcal {K} is a field K together with a finite number of automorphisms of K. This paper studies inversive extensions of inversive difference fields whose underlying field extensions are separable algebraic. The principal tool in our investigations is a Galois theory, first developed by A. E. Babbitt, Jr. for finite dimensional extensions of ordinary difference fields and extended in this work to partial difference field extensions whose underlying field extensions are infinite dimensional Galois. It is shown that if L \mathcal {L} is a finitely generated separable algebraic inversive extension of an inversive partial difference field K \mathcal {K} and the automorphisms of K \mathcal {K} commute on the underlying field of K \mathcal {K} then every inversive subextension of L / K \mathcal {L}/\mathcal {K} is finitely generated. For ordinary difference fields the paper makes a study of the structure of benign extensions, the group of difference automorphisms of a difference field extension, and two types of extensions which play a significant role in the study of difference algebra: monadic extensions (difference field extensions L / K \mathcal {L}/\mathcal {K} having at most one difference isomorphism into any extension of K \mathcal {K} ) and incompatible extensions (extensions L / K , M / K \mathcal {L}/\mathcal {K},\mathcal {M}/\mathcal {K} having no difference field compositum).

in B. The classical method of field extension (forming the polynomial ring A [x] and reducing modulo the principal ideal J of 1.1) solves this problem. In this paper, we investigate the role of this process in topological rings. We study the problem of obtaining a normed linear algebra extension B containing a solution of 1.1, where A is a given normed linear algebra. Our method is to define a norm first in the polynomial algebra A [x], and to do it in such a way that J is closed (for otherwise the canonical norm in

The last comparison is that of H*(H, A) with the Amitsur cohomology of A. First we show that there always is a natural transformation from the Amitsur cohomology of A to H*(H, A). We then specialize to the case that A is a finite field extension and give conditions on A and H which imply the natural transformation is an isomorphism. We also show that many field extensions A can

We show that a valued field A with only a finite number of nonisomorphic valued extensions is equal to the complex field C or is real closed with C=A(x/(-1)). The Ostrowski (Gelfand-Mazur) Theorem [2, p. 131], [4, p. 260] implies that with any of the valuations v(x) = I xj t, where j * I denotes the usual modulus and 0

Throughout this note L/K denotes a purely inseparable field extension of characteristic p and nonzero exponent. In [5, p. 745], Rygg proves that when L/K has bounded exponent, then a subset M of L is a relative ?-base of L/K if and only if M is a minimal generating set of L/K. The purpose of this note is to answer the following question : If every relative ?-base of L/K is a minimal generating set, then must L/K he of bounded exponent? The answer is known to be yes when K and 2>* are linearly disjoint, * = 1, 2, • • • , see [l]. We give two examples for which the answer is no: One in which the maximal perfect subfield of L is contained in K, and the other in which it is not. The following lemmas are needed for our examples. An intermediate field V of L/K is called proper if KCLL'EL.

1. Let F be an extension field of K. A minimal set of generators of F over K is a subset S of F such that F=K(S) and S'CS implies K(S') CK(S) where C denotes proper inclusion. Pickert [4, p. 881 has shown that if F is a finite inseparable extension of K (the characteristic of K is p O) and S= {a,, * * *, an} is a minimal set of generators of F over K, then S is p-independent in F (this concept, due to Teichniiller [5], is defined in ?2 following) and is a minimal set of generators of F over FP(K). A relative p-basis of F over K, as introduced in [5], is a minimal set of generators of F over FP(K). It is shown by Becker and MacLane [i, Theorem 6] that if F is a finite purely inseparable extension of K, then the minimal number of generators of F over K is n, the exponent determined by the degree [F: FP(K) ] = pn. Closely related results are given by Weil [6, Chapter I, ?5] and by Zariski and Samuel [7, Chapter II, ?171 in a discussion of derivations on fields. In this note we assume that F is a purely inseparable extension of K of arbitrary degree but with finite exponent e: FP CK. It is the purpose of this note to prove the following

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

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.