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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

We prove a relative trace formula that establishes the cubic base change for GL(2). One also gets a classification of the image of base change. The case when the field extension is nonnormal gives an example where a trace formula is used to prove lifting which is not endoscopic.

Let $Aut_kK$ be the automorphism scheme of a finite purely inseparable field extension $k\to K$ (in fact, we consider a wide class of finite ring extensions). Let $A_M$ be the splitting algebra of K (see 2.8) and $K_M={A_M}_{\text red} $ the splitting field of K. It is proved that $Aut_kK$ is integral if and only if $A_M=K_M$ . This may be formulated as a condition on the degree of $K_M$ and generalizes a result of Chase. Surprisingly, $A_M$ may be defined intrinsically, since $\mathrm{Spec} A_M$ is the scheme parametrizing the maximal smooth subgroups of $Aut_kK$ . It is also proved that the desingularization of $Aut_kK$ is the universal maximal smooth subgroup of $Aut_kK$ and coincides with the blowing up along a closed subscheme canonically defined from the action of $Aut_kK$ on $A_M$ .

Let LF be a finite separable field extension of degree n, X a smooth quasi-projective L-scheme, and R(X) the Weil transfer of X with respect to LF. The map Z R(Z) of the set of simple cycles Z ⊂ X extends in a natural way to a map Z(X) → Z(R(X)) on the whole group of algebraic cycles Z(X). This map factors through the rational equivalence of cycles and induces this way a map of the Chow groups CH(X) → CH(R(X)), which, in its turn, produces a natural functor of the categories of Chow correspondences CV(L) → CV(F). Restricting to the graded components, one has a map Z∗(X) → Zn · ∗(R(X)), which produces a functor of the categories of degree 0 Chow correspondences CV0(L) → CV0(F), a functor of the categories of the Grothendieck Chow-motivesM(L) → M(F), as well as functors of several other classical motivic categories.

This paper describes both the implementation and the mathematics of an Abstract Algebra project. The project begins with a seemingly simple question about a polynomial equation. It provides a concrete setting for exploring the relationships between polynomials, matrices, modular arithmetic, unique factorization, roots of unity, and order of group elements to help students understand the underlying abstract ring, field, and group theory.

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.

A field extension L / F is called excellent if, for any quadratic form φ over F, the anisotropic part (φL)an of φ over L is defined over F; L / F is called universally excellent if L ⋅ E / E is excellent for any field extension E / F. We study the excellence property for a generic splitting field of a central simple F-algebra. In particular, we show that it is universally excellent if and only if the Schur index of the algebra is not divisible by 4. We begin by studying the torsion in the second Chow group of products of Severi–Brauer varieties and its relationship with the relative Galois cohomology group H3(L / F) for a generic (common) splitting field L of the corresponding central simple F-algebras.

Letpbe a prime number. LetMbe a finite Galois extension of a finite algebraic number fieldk. Suppose thatMcontains a primitive pth root of unity and that thep-Sylow subgroup of the Galois groupG= Gal(M/k) is normal. LetKbe the intermediate field corresponding to thep-Sylow subgroup. Let= Gal(K/k). Thep-class groupCofMis a module over the group ringZpG, whereZpis the ring ofp-adic integers. LetJbe the Jacobson radical ofZpG. C/JCis a module over a semisimple artinian ringFp. We study multiplicity of an irreducible representation Φ apperaring inC/JCand prove a formula giving this multiplicity partially. As application to this formula, we study a cyclotomic fieldMsuch that the minus part ofCis cyclic as aZpG-module and a CM-fieldMsuch that the plus part ofCvanishes for oddp.To show the formula, we apply theory of central extensions of algebraic number field and study global and local Kummer duality between the genus group and the Kummer radical for the genus field with respect toM/K.

A fieldK is called stable if every finitely generaed regular field extensionF/K has a transcendence basex1, …,xn with the following properties: The field extensionF/K(x1,…,xn) is separable and the Galois hull\(\hat F\) ofF/K(x1,…,xn) remains regular overK, i.e.K is algebraically closed in\(\hat F\). We prove in this paper thatevery field is stable. This generalizes results from [FJ1] and [GJ] which prove that fields of characteristic 0 and infinite perfect fields are stable, respectively. [G] showed that finite fields are stable in dimension 1, i.e. every finitely generated regular field extension of transcendence degree 1 over a finite field has a stable transcendence base. In the last section of this paper we apply the theorem to the construction of PAC fields with additional properties. A fieldK is called PAC if every absolutely irreducible variety overK has at least oneK-rational point.

Let φ be an anisotropic quadratic form over a field F of characteristic not 2. The splitting pattern of φ is defined to be the increasing sequence of nonnegative integers obtained by considering the Witt indices iW(φk) of φ over K where K ranges over all field extensions of F. Restating earlier results by HURRELBRINK and REHMANN, we show how the index of the Clifford algebra of φ influences the splitting pattern. In the case where F is formally real, we investigate how the signatures of φ influence the splitting behaviour. This enables us to construct certain splitting patterns which have been known to exist, but now over much “simpler” fields like formally real global fields or ℝ(t). We also give a full classification of splitting patterns of forms of dimension less than or equal to 9 in terms of properties of the determinant and Clifford invariant. Partial results for splitting patterns in dimensions 10 and 11 are also provided. Finally, we consider two anisotropic forms φ and φ of the same dimension m with φ ⟂ − φ ∈ InF and give some bounds on m depending on n which assure that they have the same splitting pattern.

Let K be F_q((T)), or more generally any field of characteristic p equipped with a valuation having a finite residue field of q elements. Then a polynomial f(x) in K[x] having k+1 nonzero coefficients has at most q^k distinct zeros in K. We also obtain the best possible bound for the number of zeros of degree at most d over K., 6 pages, Latex 2e; to appear in Math. Res. Letters

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

We continue the work of Kaplansky on immediate valued field extensions and determine special properties of elements in such extensions. In particular, we are interested in the question when an immediate valued function field of transcendence degree 1 is henselian rational (i.e., generated, modulo henselization, by one element). If so, then wild ramification can be eliminated in this valued function field. The results presented in this paper are crucial for the first author's proof of henselian rationality over tame fields, which in turn is used in his work on local uniformization.

To what extent does the maximal subfield spectrum of a division algebra determine the isomorphism class of that algebra? It has been shown that over some fields a quaternion division algebra's isomorphism class is largely if not entirely determined by its maximal subfield spectrum. However in this paper, we show that there are fields for which the maximal subfield spectrum says little to nothing about a quaternion division algebra's isomorphism class. We give an explicit construction of a division algebra with infinite genus. Along the way we introduce the notion of a "linking field extension," which we hope will be of independent interest. We go on to show that there exists a field K for which (1) there are infinitely many nonisomorphic quaternion division algebras with center K, and (2) any two quaternion division algebra with center K are pairwise weakly isomorphic. In fact we show that there are infinitely many nonisomorphic fields satisfying these two conditions., 5 pages. In new version, title is changed from "A Division Algebra With Infinite Genus" to match that of published version

Let $p$ be a prime integer and $F$ a field of characteristic 0. Let $X$ be the {\em norm variety} of a symbol in the Galois cohomology group $H^{n+1}(F,��_p^{\otimes n})$ (for some $n\geq1$), constructed in the proof of the Bloch-Kato conjecture. The main result of the paper affirms that the function field $F(X)$ has the following property: for any equidimensional variety $Y$, the change of field homomorphism $\CH(Y)\to\CH(Y_{F(X)})$ of Chow groups with coefficients in integers localized at $p$ is surjective in codimensions $< (\dim X)/(p-1)$. One of the main ingredients of the proof is a computation of Chow groups of a (generalized) Rost motive (a variant of the main result not relying on this is given in Appendix). Another important ingredient is {\em $A$-triviality} of $X$, the property saying that the degree homomorphism on $\CH_0(X_L)$ is injective for any field extension $L/F$ with $X(L)\ne\emptyset$. The proof involves the theory of rational correspondences reviewed in Appendix., 38 pages; final version, to appear in Ann. Sci. \'Ec. Norm. Sup\'er. (4)

In this paper, we study the fine Selmer groups attached to a Galois module defined over a commutative complete Noetherian ring with finite residue field of characteristic p. Namely, we are interested in its properties upon taking residual representation and within field extensions. In particular, we will show that the variation of the fine Selmer group in a cyclotomic $\Zp$-extension is intimately related to the variation of the class groups in the cyclotomic tower. We also discuss some examples of pseudo-nullity of fine Selmer groups., Comment: 31 pages, numerous corrections and changes

LetK be the field of fractions of a curve overR whereR is the henselization of a regular local ring on an algebraic curve over a field which is algebraically closed and has characteristic 0. ThenK has the exponent=degree property for division algebras. In fact every central finite dimensionalK-division algebra with exponentn is a cyclic algebra of degreen.

The study of extensions with prescribed ramification plays a central role in algebraic number theory. It is a general hope to benefit from the analogy between number fields and function fields of trancendency degree 1 over finite fields, because for the latter the situation is much better understood. In this paper we introduce a new kind of restriction for the ramification of primes in extensions of algebraic number fields and we study the corresponding covering type over the ring of integers. We will prove that the associated fundamental group has a description similar to that of the 6tale fundamental group of a smooth, projective curve over a finite field. Let K be a number field, p be a prime number and let S be a finite set of places of K containing all archimedian places and all places dividing p. Extensions of K which are unramified outside S correspond to &ale covers of the open subscheme Spec(Ox, s) of Spec(Cx). We denote the maximal pro-p Galois extension of K which is unramified outside S by Ks(p) and we write

Let K/Kobe a finitely generated field extension of transcendence degree 1. Let u0 be a valuation of Koand v a valuation of Kextending v0such that the residue field of vis a transcendental extension ofthe residue field k0of vo/such a prolongation vwill be called a residually transcendental prolongation of v0. Byan element with the uniqueness propertyfor (K, v)/(K0, v0) (or more briefly for v/v0)we mean an element / of Khaving u-valuation 0 which satisfies (i) the image of tunder the canonicalhomomorphism from the valuation ring of vonto the residue field of v(henceforth referred to as the v-residue ot t) is transcendental over ko; that is vcoincides with the Gaussian valuation on the subfield K0(t) defined by (ii) vis the only valuation of K (up to equivalence) extending the valuation .

Letk be an algebraically closed field, and letR be a finitely generated, connected gradedk-algebra, which is a domain of Gelfand-Kirillov dimension two. Write the graded quotient ringQ(R) ofR asD[z,z−1; δ], for some automorphism δ of the division ringD. We prove thatD is a finitely generated field extension ofk of transcendence degree one. Moreover, we describeR in terms of geometric data. IfR is generated in degree one then up to a finite dimensional vector space,R is isomorphic to the twisted homogeneous coordinate ring of an invertible sheaf ℒ over a projective curveY. This implies, in particular, thatR is Noetherian, thatR is primitive when |δ|=∞ and thatR is a finite module over its centre when |δ

The Gauss map of a projective variety $X \subset \mathbb{P}^N$ is a rational map from $X$ to a Grassmann variety. In positive characteristic, we show the following results. (1) For given projective varieties $F$ and $Y$, we construct a projective variety $X$ whose Gauss map has $F$ as its general fiber and has $Y$ as its image. More generally, we give such construction for families of varieties over $Y$ instead of fixed $F$. (2) At least in the case when the characteristic is not equal to $2$, any inseparable field extension appears as the extension induced from the Gauss map of some $X$., Comment: 27 pages, Corollary 5.4 added

We give a purely algebraic construction of the continuous closure of any finitely generated torsion free module; a concept first studied by H.~Brenner and M.~Hochster. The construction implies that, at least in characteristic 0, taking continuous closure commutes with flat morphisms whose fibers are semi-normal. This implies that the continuous closure of a coherent ideal sheaf is again a coherent ideal sheaf (both in the Zariski and in the \'etale topologies) and it commutes with field extensions.

In [BGLM] and [GLNP] it was conjectured that if $H$ is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in $H$ of covolume at most $x$ is $x^{(\gamma(H)+o(1))\log x/\log\log x}$ where $\gamma(H)$ is an explicit constant computable from the (absolute) root system of $H$. In this paper we prove that this conjecture is false. In fact, we show that the growth is at rate $x^{c\log x}$. A crucial ingredient of the proof is the existence of towers of field extensions with bounded root discriminant which follows from the seminal work of Golod and Shafarevich on class field towers., Comment: 27 pages, a small change in title, final revision, to appear in Adv. Math

The paper has three main applications. The first one is this Hilbert-Grunwald statement. If $f:X\rightarrow \Pp^1$ is a degree $n$ $\Qq$-cover with monodromy group $S_n$ over $\bar\Qq$, and finitely many suitably big primes $p$ are given with partitions $\{d_{p,1},..., d_{p,s_p}\}$ of $n$, there exist infinitely many specializations of $f$ at points $t_0\in \Qq$ that are degree $n$ field extensions with residue degrees $d_{p,1},..., d_{p,s_p}$ at each prescribed prime $p$. The second one provides a description of the se-pa-ra-ble closure of a PAC field $k$ of characteristic $p\not=2$: it is generated by all elements $y$ such that $y^m-y\in k$ for some $m\geq 2$. The third one involves Hurwitz moduli spaces and concerns fields of definition of covers. A common tool is a criterion for an \'etale algebra $\prod_lE_l/k$ over a field $k$ to be the specialization of a $k$-cover $f:X\rightarrow B$ at some point $t_0\in B(k)$. The question is reduced to finding $k$-rational points on a certain $k$-variety, and then studied over the various fields $k$ of our applications.

We consider the question of when a semigroup is the semigroup of a valuation dominating a two dimensional noetherian local domain, giving some surprising examples. We give a necessary and sufficient condition for the pair of a semigroup S and a field extension L/k to be the semigroup and residue field of a valaution dominating a regular local ring R of dimension two with residue field k, generalizing the theorem of Spivakovsky for the case when there is no residue field extension., 34 pages. Final version

A differential version of the classical theorem of Campbell is presented In this paper we establish the equivalence of the two existing definitions of Picard–Vessiot extension of partial differential fields and give an equivalent statement of the Jacobian conjecture in terms of Picard–Vessiot extensions. We denote as usual by K〈S〉 the differential field differentially generated by the set S, and by CK the field of constants of a differential field K. Kolchin gave in [K] a definition of Picard–Vessiot extension for partial differential fields which is a generalization of the definition of Picard–Vessiot extension associated with a linear differential operator defined over an ordinary differential field. With a partial differential field extension L|K, with derivations ∂1, . . . , ∂m, he associates an ordinary differential field extension LD|KD, where KD := K〈u1, . . . , um〉, LD := L〈u1, . . . , um〉, with u1, . . . , um independent differential indeterminates, and the derivation is D := u1∂1 + · · ·+ um∂m. Received November 18, 2009

Letq be a power of 2 at least equal to 8 and ζ be a primitiveq-th root of unity, and letK be any field of characteristic zero. We define the group of special projective conormsS K as a quotient of the group of elements ofK(ζ) of norm 1:S K is obviously trival if the groul Gal (K(ζ)/K) is cyclic. We prove that for some fieldsK, the groupS K is finite, and it is even trivial for certain fields such as ℚ or ℚ(X 1,...,X m). We then prove that the groupS K completely paramatrizes the cycle extensions ofK of degreeq. We exhibit an explicit polynomial defined over ℚ(T 0,...,T q/2) which parametrizes all cyclic extensions ofK of degreeq associated to the trivial element ofS K. In particular, this polynomial parametrizes all cyclic extensions ofK of degreeq whenever the groupS K is trivial.

We show how the analogues of Jacobi sums, in the context of function fields, introduced and studied in [T1, T2, T3] can be obtained from shtukas introduced and studied in [D2, D3, M]. We apply this to obtain some results on the prime factorization of analogues of Gauss sums and to prove an analogue of the Gross-Koblitz formula for general function field, generalizing the results in [T2]. For this purpose, we also introduce and interpolate a new analogue of gamma function.

We determine the Zariski-dense subgroups of Chevalley groups and their twisted analogues over infinite algebraic extensions of finite fields. It turns out that these are essentially forms of the same group (possibly becoming twisted) over smaller infinite fields. It follows from our classification that if\(\bar G\) is a simple algebraic group over the algebraic closure of a finite field, then a dense subgroup of\(\bar G\) can never be maximal, and so the maximal subgroups of\(\bar G\) are necessarily closed. It follows that Seitz’s determination of the closed maximal subgroups of\(\bar G\) actually gives all the maximal subgroups.

We investigate the spread arising from a field extension and its chains. The major tool in this paper is the concept of transversal lines of a chain which is closely related to the Cartan—Brauer—Hua theorem. Provided that one chain has a “sufficiently large” number of such lines, both this chain and the given spread permit a simple geometric description by means of collineations.

1. Parallel edges on polygons Certain geometrical problems may be formulated algebraically. The paradigm case is the solution of ruler and compass construction problems by means of the theory of field extensions. Yet other examples can be found. We will show here how two results on regular polygons are related to permutations of the elements of a finite abelian group. In this note, G denotes a finite abelian group written multiplicatively. In order to avoid the trivial cases we suppose that the order of G is at least two. A complete mapping of G is a permutation f of G such that

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 .

Let X be a geometrically split, geometrically irreducible variety over a field F satisfying Rost nilpotence principle. Consider a field extension E/F and a finite field K. We provide in this note a motivic tool giving sufficient conditions for so-called outer motives of direct summands of the Chow motive of X_E with coefficients in K to be lifted to the base field. This going down result has been used S. Garibaldi, V. Petrov and N. Semenov to give a complete classification of the motivic decompositions of projective homogeneous varieties of inner type E_6 and to answer a conjecture of Rost and Springer., Final version of the manuscript

Letk be an infinite field of characteristic not equal to 2, 3, 5. In this paper, we construct a natural map from the set of orbits of certain prehomogeneous vector spaces to the set of isomorphism classes of Galois extensions ofk which are splitting fields of equations of certain degrees, and prove that the inverse image of this map corresponds bijectively with conjugacy classes of Galois homomorphisms.

LetR⊂T be domains, not fields, such that Spec(R)=Spec(T) as sets; that is, such that the prime ideals ofT coincide, as sets, with those ofR. It is proved that the canonical map Spec(T[[X]])→Spec(R[[X]]) is a homeomorphism. This generalizes a result of Girolami in caseR is a pseudovaluation domain with the SFT (strong finite type)—property andT is its associated valuation domain. The analogous property for polynomial rings is also characterized: Spec(T[X])→Spec(R[X]) is a homeomorphism if and only ifR/M⊂T/M is a purely inseparable (algebraic) field extension, whereM is the maximal ideal ofR.

Given a field F, one may ask which finite groups are Galois groups of field extensions E/F such that E is a maximal subfield of a division algebra with center F. This question was originally posed by Schacher, who gave partial results over the field of rational numbers. Using patching, we give a complete characterization of such groups in the case that F is the function field of a curve over a complete discretely valued field with algebraically closed residue field of characteristic zero, as well as results in related cases., 20 pages. In Section 3 some statements were strengthened and proofs simplified. At the end of Section 4 the definition of the rank two valuation was fixed

This article presents a theory of modules with iterative connection. This theory is a generalisation of the theory of modules with connection in characteristic zero to modules over rings of arbitrary characteristic. We show that these modules with iterative connection (and also the modules with integrable iterative connection) form a Tannakian category, assuming some nice properties for the underlying ring, and we show how this generalises to modules over schemes. We also relate these notions to stratifications on modules, as introduced by A. Grothendieck in order to extend integrable (ordinary) connections to finite characteristic. Over smooth rings, we obtain an equivalence of stratifications and integrable iterative connections. Furthermore, over a regular ring in positive characteristic, we show that the category of modules with integrable iterative connection is also equivalent to the category of flat bundles as defined by D. Gieseker. In the second part of this article, we set up a Picard-Vessiot theory for fields of solutions. For such a Picard-Vessiot extension, we obtain a Galois correspondence, which takes into account even nonreduced closed subgroup schemes of the Galois group scheme on one hand and inseparable intermediate extensions of the Picard-Vessiot extension on the other hand. Finally, we compare our Galois theory with the Galois theory for purely inseparable field extensions., 37 pages; v2->v3: more cross references to other papers are added in this version, the introduction is more detailed v3->v4: proof of Thm. 11.5iv) and of the following corollaries changed and hyperref added

Let W be an associative PI algebra over a field F of characteristic zero, graded by a finite group G. Let id_{G}(W) denote the T-ideal of G-graded identities of W. We prove: 1. {[G-graded PI equivalence]} There exists a field extension K of F and a finite dimensional Z/2ZxG-graded algebra A over K such that id_{G}(W)=id_{G}(A^{*}) where A^{*} is the Grassmann envelope of A. 2. {[G-graded Specht problem]} The T-ideal id_{G}(W) is finitely generated as a T-ideal. 3. {[G-graded PI-equivalence for affine algebras]} Let W be a G-graded affine algebra over F. Then there exists a field extension K of F and a finite dimensional algebra A over K such that id_{G}(W)=id_{G}(A)., Comment: 37 pages

Let K 0 (x) be a simple transcendental extension of a field K 0 , υ 0 be a valuation of K 0 with value group G 0 and residue field K 0 . Suppose G 0 ⊆G 1 ⊆G is an inclusion of totally ordered abelian groups with [G 1 : G 0 ] 1 and an infinite cyclic group. It is proved that there exists an (explicitly constructible) valuation υ of K 0 (x) extending υ 0 such that the value group of υ is G and its residue field is k, where k is a given finite extension of k 0 . This is analogous to a result of Matignon and Ohm [2, Corollary 3.2] for residually non-algebraic prolongations of υ 0 to K 0 (x).

LetF ⊂K be a field extension,A be aK-algebra. It is proved that, in general, GK dimFA≥GK dimKA+trF(K). For commutative algebras or Noetherian P.I. algebras, the equality holds. Two examples are also constructed to show that: (i) there exists an algebraA such that GK dimFA=GK dimKA+trF(K)+1; (ii) there exists an algebraic extensionF ⊂K and aK-algebraA such that GK dimFA=∞, but GK dimKA

Pfister forms over fields are those anisotropic forms that remain round under any field extension. Here, round means that for any represented element x Φ 0 the isometry xφ = φ holds where φ is the form under consideration. We investigate whether a similar characterization can be given for the round forms themselves. We obtain several "going-up" and "going-down" theorems. Some counter- examples are given which show that a general theorem holds neither in the going-up nor in the going-down situation. Introduction. Pfister forms over fields can be characterized as those anisotropic forms that remain round under any field extension (cf. (16, p. 153)). In this paper we investigate whether a similar characteriza- tion can be given for round forms over fields. Since the structure of round forms is not known in general, one cannot expect general results. According to a theorem of Marshall (14) (see also Becker/Kόpping (3)), every round form has a decomposition φ = I x ψ + p where ψ is a Pfister form and p is torsion. This implies that any round form remains round over the Pythagorean closure of the underlying field. In this paper we shall prove the following characterization theorem for certain types of forms or fields: A form φ over F is round iff it is round over every proper quadratic extension K = F(y/w) where w is represented by φ over F. As to the going-down part of this equiva- lence the usual techniques (norm principles) allow one to prove many results. The going-up part, however, requires detailed information on the structure of round forms which is available only for certain classes of fields, for example the linked fields. Counter-examples show that in general neither implication of the equivalence is true. For extensions of odd dimension we have the well-known theorem of Springer which yields immediately that a form over F which is round over K > F ((K : F) e 2N + 3) is also round over F. In the other direction, nothing is known. We use the standard terminology as is found in (16). The fields occurring in this paper are commutative and of characteristi c Φ 2. K usually denotes a field, W{K) the corresponding Witt ring, and

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)

The number of Galois extensions, up to isomorphism, of a local field whose Galois groups are isomorphic to the modular group $M_{p^{m}}=\langle x,y\mid x^{p^{m-1}}=y^{p}=1,y^{-1}xy=x^{p^{m-2}+1}\rangle$, where $p$ is an odd prime, is counted.

In this paper we compute the Fourier spectra of some recently discovered binomial APN functions. One consequence of this is the determination of the nonlinearity of the functions, which measures their resistance to linear cryptanalysis. Another consequence is that certain error-correcting codes related to these functions have the same weight distribution as the 2-error-correcting BCH code. Furthermore, for fields of odd degree, our results provide an alternative proof of the APN property of the functions., 20 pages. Submitted to the SIAM Journal on Discrete Mathematics

We introduce the condition of a profinite group being semi-free, which is more general than being free and more restrictive than being quasi-free. In particular, every projective semi-free profinite group is free. We prove that the usual permanence properties of free groups carry over to semi-free groups. Using this, we conclude that if k is a separably closed field, then many field extensions of k((x,y)) have free absolute Galois groups., Comment: 24 pages