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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.

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) .

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 .

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 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 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.

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.

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

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].

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

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

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.

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.

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} .

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.

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 ⊇ 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

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.

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

Let K be an Nk-generated field extension of the field F with transcendence degree n. Set bidim(K)=the projective dimension of K as a K OF K-module. Then K9ocally separably generated implies bidim(K)

1. L. S. Bosanquet, A mean value theorem, J. London Math. Soc. vol. 16 (1941) pp. 146-148. 2. G. H. Hardy, Divergent series, Oxford, 1949. 3. W. Jurkat, Uber Rieszsche Mittel mit unstetigem Parameter, Math. Z. vol. 55 (1951) pp. 8-12. 4. W. Jurkat and A. Peyerimhoff, Mittelwertsdtze bei Matrixund Integraltransformationen, Math. Z. vol. 55 (1951) pp. 92-108. 5. -, Mittelwertsdtze und Vergleichssdtze fiir Matrixtransformationen, Math. Z. vol. 56 (1952) pp. 152-178. 6. G. G. Lorentz, Direct theorems on methods of summability, Canad. J. Math, vol. 1 (1949) pp. 305-319. 7. I. Schur, Uber lineare Transformationen in der Theorie der unendlichen Reihen, J. Reine Angew. Math. vol. 151 (1921) pp. 79-111. 8. A. Wilansky, A necessary and sufficient condition that a summability method be stronger than convergence, Bull. Amer. Math. Soc. vol. 55 (1949) pp. 914-916.

Let F be any field and suppose that E is a separable algebraic extension of F. For elements aGE, we let dga denote the degree of the minimal polynomial of a over F. Let a, O E, dga=m, dgo3=n and suppose (m, n) = 1. It is easy to see that [F(a, ,B): F] =nn, and by a standard theorem of field theory (for instance see Theorem 40 on p. 49 of [I]), there exists an element -y E such that F(a, r) -F(,y) and thus dgy = in. In fact, the usual proof of this theorem produces (for infinite F) an element of the form zy =-a +X3, with X E F. In this paper we show that in many cases the choice of X EF is completely arbitrary, as long as X#zO. In Theorem 63 on p. 71 of [1], it is shown that if n>m and n is a prime different from the characteristic of F, then dg(a + =rmn. The present result includes this.