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For a given field F of characteristic different from 2 and $$a,b,d\in F^*$$ we construct an invariant $$\mathrm{inv}$$ for an element $$D\in \,_2\mathrm{Br}(F(\sqrt{a},\sqrt{b},\sqrt{d})/F)$$ . This invariant takes value in the quotient group $$\begin{aligned} H^3(F,\mu _2)/D\cup {\mathrm{N}_{\mathrm{F}\left( \sqrt{\mathrm{d}}, \sqrt{\mathrm{ab}}\right) /\mathrm{F}}}F\left( \sqrt{d},\sqrt{ab}\right) ^*. \end{aligned}$$ Let k be a field, let $$k(\sqrt{a},\sqrt{b},\sqrt{d})/k$$ be a triquadratic field extension. We apply the invariant $$\mathrm{inv}$$ and a few deep results from algebraic geometry and K-theory to construct a field extension K/k with $$\mathrm{cd}_2 K=3$$ , and an indecomposable cross product algebra of exponent 2 with respect to the extension $$K(\sqrt{a},\sqrt{b},\sqrt{d})/K$$ . Using the invariant $$\mathrm{inv}$$ , we also prove the following odd degree descent statement: Assume $$D\in \,_2\mathrm{Br}(F)$$ , $$b,d\in F^*$$ , L/F is an odd degree extension. Assume also that $$D_{L(\sqrt{b},\sqrt{d})}=Q_{L(\sqrt{b},\sqrt{d})}$$ , where Q is a quaternion algebra defined over L. Then there exists a quaternion algebra $$\widetilde{Q}$$ defined over F such that $$D_{F(\sqrt{b},\sqrt{d})}=\widetilde{Q}_{F(\sqrt{b},\sqrt{d})}$$ . As a consequence we get that if $$\phi \in I^2(F)$$ is a form such that $${(\phi _{L(\sqrt{b},\sqrt{d})})}_{an}$$ is defined over L, and $$\dim {(\phi _{L(\sqrt{b},\sqrt{d})})}_{an} =4$$ , then $${(\phi _{F(\sqrt{b},\sqrt{d})})}_{an}$$ is defined over F.

Let $k$ be a field with char $k\neq 2$ and $k$ be not algebraically closed. Let $a\in k\setminus k^2$ and $L=k(\sqrt{a})(x,y)$ be a field extension of $k$ where $x,y$ are algebraically independent over $k$. Assume that $\sigma$ is a $k$-automorphism on $L$ defined by \[ \sigma: \sqrt{a}\mapsto -\sqrt{a},\ x\mapsto \frac{b}{x},\ y\mapsto \frac{c(x+\frac{b}{x})+d}{y} \] where $b,c,d \in k$, $b\neq 0$ and at least one of $c,d$ is non-zero. Let $L^{\langle\sigma\rangle}=\{u\in L:\sigma(u)=u\}$ be the fixed subfield of $L$. We show that $L^{\langle\sigma\rangle}$ is isomorphic to the function field of a certain surface in $P^4_k$ which is given as the intersection of two quadrics. We give criteria for the $k$-rationality of $L^{\langle\sigma\rangle}$ by using the Hilbert symbol. As an appendix of the paper, we also give an alternative geometric proof of a part of the result which is provided to the authors by J.-L. Colliot-Th\'el\`ene., Comment: To appear in Manuscripta Math. The main theorems (old Theorem 1.7 and Theorem 1.8) incorporated into (new) Theorem 1.8. Section 3 and Section 4 interchanged

Let K be a finitely generated extension of a field k of characteristic $$p\not =0$$ . In 1947, Dieudonne initiated the study of maximal separable intermediate fields. He gave in particular the form of an important subclass of maximal separable intermediate fields D characterized by the property $$K\subseteq k({D}^{p^{-\infty }})$$ , and which are called the distinguished subfields of K/k. In 1970, Kraft showed that the distinguished maximal separable subfields are precisely those over which K is of minimal degree. This paper grew out of an attempt to find a new characterization of distinguished subfields of K/k by means of new inseparability invariants.

Over a global field any finite number of central simple algebras of exponent dividing m is split by a common cyclic field extension of degree m. We show that the same property holds for function fields of 2-dimensional excellent schemes over a henselian local domain of dimension one or two with algebraically closed residue field.

We initiate the study of affine Deligne–Lusztig varieties with arbitrarily deep level structure for general reductive groups over local fields. We prove that for $${{\,\mathrm{GL}\,}}_n$$ and its inner forms, Lusztig’s semi-infinite Deligne–Lusztig construction is isomorphic to an affine Deligne–Lusztig variety at infinite level. We prove that their homology groups give geometric realizations of the local Langlands and Jacquet–Langlands correspondences in the setting that the Weil parameter is induced from a character of an unramified field extension. In particular, we resolve Lusztig’s 1979 conjecture in this setting for minimal admissible characters.

Let k be a field of characteristic p≠ 0. In 1968, M. E. Sweedler revealed for the first time, the usefulness of the concept of modularity. This notion, which plays an important role especially for Galois theory of purely inseparable extensions, was used to characterize purely inseparable extensions of bounded exponent which were tensor products of simple extensions. A natural extension of the definition of modularity is to say that K/k is q-modular (quasi-modular) if K is modular up to some finite extension. In subsequent papers, M. Chellali and the author have studied various property of q-modular field extensions, including the questions of q-modularity preservation in case [k : kp] is finite. This paper grew out of an attempt to find analogue results concerning the preservation of q-modularity, without the hypothesis on k but with extra assumptions on K/k. In particular, we investigate existence conditions of lower (resp. upper) quasi-modular closures for a given q-finite extension.

We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal $L$-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal $L$-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of $L$-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$, applicable to infinite families of fields of arbitrarily large degree., Comment: 52 pages. This shorter version aligns with the published paper. Note that portions of Section 8 of the longer v1 have been developed as a separate paper with identifier arXiv:1902.02008

A theorem of Chow concerns homomorphisms of two abelian varieties under a primary field extension base change. In this paper, we generalize Chow’s theorem to semi-abelian varieties. This contributes to different proofs of a well-known result that every algebraic torus splits over a finite separable field extension. We also obtain the best bound for the degrees of splitting fields of tori.

An 1861 theorem of Hermite asserts that for every field extension E / F of degree 5 there exists an element of E whose minimal polynomial over F is of the form $$f(x) = x^5 + c_2 x^3 + c_4 x + c_5$$ for some $$c_2, c_4, c_5 \in F$$ . We give a new proof of this theorem using techniques of Galois cohomology, under a mild assumption on F.

We develop a modification of a notion of distance of an element in a valued field extension introduced by F.-V. Kuhlmann. We show that the new notion preserves the main properties of the distance and at the same time gives more complete information about a valued field extension. We study valued field extensions of prime degree to show the relation between the distances of the elements and the corresponding extensions of value groups and residue fields. In connection with questions related to defect extensions of valued function fields of positive characteristic, we present constructions of defect extensions of rational function fields K(x, y)|K generated by elements of various distances from K(x, y). In particular, we construct dependent Artin–Schreier defect extensions of K(x, y) of various distances.

Let k be a field of characteristic $$ p\not =0 $$. For a (purely) inseparable extension K / k the notion of modularity, defined by M.E. Sweedler in the 60s, is a very important property, very much like being Galois for a separable extension. We have defined, together with M. Chellali, a generalization of the notion of modularity, called lower quasi-modularity: K / k is lower quasi-modular (lq-modular) if for some finite extension $$k'$$ over k we have that $$K/k'$$ is modular. In subsequent papers M. Chellali and the author have studied various properties of lq-modular field extensions, including the existence of lq-modular closures in case $$[k{:}k^p]$$ is finite. In this paper we prove a similar result, without the hypothesis on k but with extra assumptions on K / k: the extension needs to be q-finite, that is, there must exist an integer M such that for every positive integer n the field $$K\cap k^{p^{-n}}$$ is generated by at most M elements on k. A number of properties of lq-modular closures are determined and examples are presented illustrating the results.

The aim of the paper is to construct an embedding of a given cyclic extension of degree p of a complete discrete valuation field of characteristic 0 with an arbitrary residue field of characteristic p > 0 into a cyclic extension of degree p2. The result extends the construction obtained by S. V. Vostokov and I. B. Zhukov in terms of Witt vectors, to a wider interval of values for the ramification jump of the original field extension.

We analyze the rel leaves of the Arnoux-Yoccoz translation surfaces. We show that for any genus $g \geq 3$, the leaf is dense in the connected component of the stratum $H(g -1 , g -1)$ to which it belongs, and the one-sided imaginary-rel trajectory of the surface is divergent. For one surface on this trajectory, namely the Arnoux-Yoccoz surface itself, the horizontal foliation is invariant under a pseudo-Anosov map (and in particular is uniquely ergodic), but for all other surfaces, the horizontal foliation is completely periodic. The appendix proves a field theoretic result needed for denseness of the leaf: for any $n \geq 3$, the field extension of the rationals obtained by adjoining a root of $X^n-X^{n-1}-\ldots-X-1$ has no totally real subfields other than the rationals., Comment: Appendix by Lior Bary-Soroker, Mark Shusterman and Umberto Zannier. The prior version was published, but had errors in \S 6. Erroneous statements have been indicated and an erratum was included as Appendix B which corrects the errors. Main results are still correct. 70 pages, 9 figures. arXiv admin note: text overlap with arXiv:1506.06773

Using a geometric approach involving Riemann surface orbifolds, we provide lower bounds for the genus of an irreducible algebraic curve of the form $$\mathcal {E}_{A,B}:\, A(x)-B(y)=0$$ , where $$A, B\in \mathbb {C}(z)$$ . We also investigate “series” of curves $$\mathcal {E}_{A,B}$$ of genus zero, where by a series we mean a family with the “same” A. We show that for a given rational function A a sequence of rational functions $$B_i$$ , such that $$\deg B_i\rightarrow \infty $$ and all the curves $$A(x)-B_i(y)=0$$ are irreducible and have genus zero, exists if and only if the Galois closure of the field extension $$\mathbb {C}(z)/\mathbb {C}(A)$$ has genus zero or one.

Let \(k_0\) be a field of characteristic distinct from 2, \(l_0{/}k_0\) a finite field extension of degree \(2^m\), \(m\ge 2\). We prove that there exists a field extension \(K{/}k_0\) linearly disjoint with \(l_0/k_0\) such that the extension \(l_0K{/}K\) is nonexcellent. The crucial point is a construction of in some sense nonstandard elements in \(H^3(K,\mathbb {Z}{/}2\mathbb {Z})\). We apply this construction as well for investigation of the group \({F^*}^2N_{L{/}F}L^*\), where L / F is a \(\mathbb {Z}{/}4\mathbb {Z}\times \mathbb {Z}{/}2\mathbb {Z}\)-Galois extension. More precisely, let \(F\subset F_i\subset L\)\((1\le i\le 3)\) be the three intermediate quadratic extensions of F, and \(N_i(F)=N_{F_i{/}F}F_i^*\). We show that the quotient group \({N_1(F)\cap N_2(F)\cap N_3(F)\over {F^*}^2N_{L/F}L^*}\) can be arbitrarily large.

In this note, we are interested in local-global principles for multinorm equations of the form $\prod_{i=1}^n N_{L_i /k}(z_i) = a$ where $k$ is a global field, $L_i/k$ are finite separable field extensions and $a \in k^*$. In particular, we prove a result relating weak approximation for this equation to weak approximation for some classical norm equation $N_{F/k}(w) = a$ where $F := \bigcap_{i=1}^n L_i$. It provides a proof of a "weak approximation" analogue of a recent conjecture by Pollio and Rapinchuk about multinorm principle. We also provide a counterexample to the original conjecture concerning Hasse principle., Comment: 14 pages, Final version

Nonassociative quaternion algebras were first discovered over the real numbers independently by Dickson and Albert and provided some of the first examples of nonassociative division algebras. They were later classified completely by Waterhouse. Cyclic algebras can be seen as a natural generalisation of the classical quaternion algebras. With this in mind we generalise nonassociative quaternion algebras and introduce nonassociative cyclic algebras. These provide new examples of nonassociative central division algebras with Nucleus a separable field extension of degree n.

The weak isotropy index (or equivalently, sublevel) of arbitrary quadratic forms is studied. Its relationship to the level of a form is investigated. The problem of determining the set of values of the weak isotropy index of a form as it ranges over field extensions is addressed, with both admissible and inadmissible numbers being determined. An analogous investigation with respect to the level of a form is also undertaken. A treatment of forms for which the above invariants coincide concludes this article, with some recently-raised questions being resolved. European Commission - Seventh Framework Programme (FP7) Irish Research Council Marie Curie Actions

Let (V, h) be a Hermitian space over a division algebra D which is of index at most two over a non-Archimedean local field k of residue characteristic not 2. Let G be the unitary group defined by h and let \({\sigma}\) be the adjoint involution. Suppose we are given two \({\sigma}\)-invariant but not \({\sigma}\)-fixed field extensions E1 and E2 of k in EndD(V) which are isomorphic under conjugation by an element g of G and suppose that there is a point x in the Bruhat–Tits building of G which is fixed by \({E_1^{\times}}\) and \({E_2^{\times}}\) in the reduced building of AutD(V). Then E1 is conjugate to E2 under an element of the stabilizer of x in G if E1 and E2 are conjugate under an element of the stabilizer of x in AutD(V) and a weak extra condition holds. In addition, in many cases the conjugation by g from E1 to E2 can be realized as conjugation by an element of the stabilizer of x in G. Further we give a concrete description of the canonical isomorphism from the set of \({E_1^\times}\) fixed points of the building of G onto the building of the centralizer of E1 in G.

We study the behavior of orderings, semiorderings, and the stability index under field extensions.

Let K/k be a separable field extension of degree 2, D be a finite-dimensional central division algebra over K with a K/k-involution τ, h be an Hermitian anisotropic form on a right D-vector space with respect to τ, and let U(h) be the unitary group of h. Then the reduced Whitehead group of its special linear subgroup is defined as follows: $$ \mathrm{SUK}_1^{\mathrm{an}}(h)={{{\mathrm{SU}(h)}} \left/ {{\left[ {U(h),U(h)} \right]}} \right.} $$ , where [U(h), U(h)] is the commutator subgroup of U(h). The first main result establishes a link between the above group and its analog SUK1(h) for the case of isotropic h (with respect to the same τ). Theorem. There exists a surjective homomorphism from $$ \mathrm{SUK}_1^{\mathrm{an}}(h) $$ to SUK1(h). Furthermore, we also give a solution of the conjugacy problem for special unitary subgroups of anisotropic Hermitian forms over quaternion division algebras as subgroups of their multiplicative groups. Bibliography: 32 titles.

Algebraic independence of certain Mahler functions constructed from Rudin–Schapiro sequences and Baum–Sweet sequences is proved, using difference Riccati equations and the notion of difference field extension of valuation ring type.

Essential dimension is an invariant of algebraic groups G over a field F that measures the complexity of G-torsors over field extensions of F. We use theorems of N. Karpenko about the incompressibility of Severi-Brauer varieties and quadratic Weil transfers of Severi-Brauer varieties to compute the essential dimension of some closed subgroups of RK/F (GL1(A)), where A is a central division K-algebra of prime power degree and K/F is a separable field extension of degree ≤ 2. In particular, we determine the essential dimension of the group Sim(A, σ) of similitudes of (A, σ), where σ is an F-involution on A, and the essential dimension of the normalizer \(N_{GL_1 (A)} \left( {GL_1 \left( B \right)} \right)\), where B is a separable subalgebra of A.

For a field F of characteristic not 2, let \({\widehat{F}}\) denote the maximal dimension of anisotropic, weakly isotropic, non-degenerate quadratic forms over F. In this paper, we investigate the behavior of this invariant under field extensions.

For a field F and a family of central simple F-algebras we prove that there exists a regular field extension E/F preserving indices of F-algebras such that all the algebras from the family are cyclic after scalar extension by E. Let \( \mathcal{A} \) be a central simple algebra over a field F of degree n with a primitive nth root of unity ρn. We construct a quasi-affine F-variety Symb(\( \mathcal{A} \)) such that for a field extension L/F Symb(\( \mathcal{A} \)) has an L-rational point if and only if \( \mathcal{A}{ \otimes_F}L \) is a symbol algebra. Let \( \mathcal{A} \) be a central simple algebra over a field F of degree n and K/F be a cyclic field extension of degree n. We construct a quasi-affine F-variety C(\( \mathcal{A} \),K) such that, for a field extension L/F with the property [KL : L] = [K : F], the variety C(\( \mathcal{A} \),K) has an L-rational point if and only if KL is a subfield of \( \mathcal{A}{ \otimes_F}L \).

Let B1 and B2 be disjoint separable algebraic extensions of a field F, and let B = B1B2 be their composite. Let α1 be an element of B1 and α2 be an element of B2. Suppose α1 and α2 are primitive (resp. normal, resp. subnormal). We investigate the question of when α1 + α2 and α1α2 are necessarily primitive (resp. normal, resp. subnormal) elements of B. (A normal element of a Galois extension is defined to be one that is part of a normal basis, and a subnormal element is defined analogously for a non-Galois extension).

Let A be a finitely generated K-algebra that is a domain of GK dimension less than 3, and let Q(A) denote the quotient division algebra of A. We show that if D is a division subalgebra of Q(A) of GK dimension at least 2, then Q(A) is finite dimensional as a left D-vector space. We use this to show that if A is a finitely generated domain of GK dimension less than 3 over an algebraically closed field K, then any division subalgebra D of Q(A) is either a finitely generated field extension of K of transcendence degree at most one, or Q(A) is finite dimensional as a left D-vector space.

In this paper, we study the algebraic structure of principal units in the tower of the maximal abelian extensions of local fields of characteristic zero and the corresponding Galois groups at each level. As an application, we show the finiteness result for the number of coverings with a given degree of the maximal abelian extension of a local field in characteristic zero. The number of p-coverings for \({\mathbb{Q}_p}\) is computed explicitly.

We study finite-dimensional representations of hyper loop algebras over non-algebraically closed fields. The main results concern the classification of the irreducible representations, the construction of the Weyl modules, base change, tensor products of irreducible and Weyl modules, and the block decomposition of the underlying abelian category. Several results are interestingly related to the study of irreducible representations of polynomial algebras and Galois theory.

Let $\cal{A}$ be a central simple algebra over a field k and G be a commutative group with $ \left| G \right| = \deg \left( \cal{A} \right) $ . It is proved that there exists a regular field extension E/k preserving indices of k-algebras such that $ {\cal{A}} \otimes _k E $ is a crossed product with the group G. Bibliography: 11 titles.

Let \(K/\mathbb {F}_q (T)\) be a finite function field extension and denote by OK the integral closure of \(\mathbb {F}_q [T]\) in K. In this article, we are interested in Polya fields, that is, fields K, such that the OK-module Int(OK) of integer-valued polynomials over OK admits a regular basis. We show that the cyclotomic extensions of \(\mathbb {F}_q (T)\) are Polya fields, and we characterize some totally imaginary extensions which are Polya fields. Then, we are interested in Polya fields K which have a regular basis of the form \(\left\{\prod_{0\le k < n}\frac{X-a_k}{a_n-a_k},\,n\in\mathbb {N}\right\}\) for some sequences \((a_n)_{n\in\mathbb {N}}\) of elements of OK. For totally imaginary extensions, we show that it is the case if and only if OK is isomorphic to \(\mathbb {F}_q [T]\). This gives a answer to a question raised by Thakur.

Towers of fields F1 ⊂ F2 ⊂ F3 are considered, where F3/F2 is a quadratic extension and F2/F1 is an extension, which is either quadratic or of odd degree or purely transcendental of degree 1. Numerous examples of the above types such that the extension F3/F1 is not 4-excellent are constructed. Also it is shown that if k is a field, char k ≠ 2, and l/k is an arbitrary field extension of fourth degree, then there exists a field extension F/k such that the fourth degree extension lF/F is not 4-excellent. Bibliography: 5 titles.

The main purpose of the paper is to strengthen previous author’s results. Let k be a field of characteristic ≠ 2, n ≥ 2. Suppose that elements \(\bar a,\overline {b_1 } , \ldots ,\overline {b_n } \in k^* /k^{*2} \) are linearly independent over ℤ/2ℤ. We construct a field extension K/k and a quaternion algebra D = (u, v) over K such that (1) the field K has no proper extension of odd degree (2) the u-invariant of K equals 4 (3) the multiquadratic extension \(K(\sqrt {b_1 } , \ldots \sqrt {b_n } )/K\) is not 4-excellent, and the quadratic form 〈uv,-u,-v, a〉 provides a relevant counterexample (4) the central division algebra A = D ⊗E (a, t0) ⊗E (b1, t1) ⋯ ⊗E (bn, tn) does not decompose into a tensor product of two nontrivial central simple algebras over E, where E = K ((t0))((t1)) … ((tn)) is the Laurent series field in the variables t0, t1, …, tn (5) ind A = 2n+1.

Let k be an algebraically closed base field of arbitrary characteristic. In this paper, we study actions of a connected solvable linear algebraic group G on a central simple algebra Q. The main result is the following: Q can be split G-equivariantly by a finite-dimensional splitting field, provided that G acts “algebraically,” i.e., provided that Q contains a G-stable order on which the action is rational. As an application, it is shown that rational torus actions on prime PI-algebras are induced by actions on commutative domains.

An element r of a field L will be called finitely arithmetically fixed (f.a.f.) if there exists some finite subset A of L containing r such that every map f from A to L which behaves “like a homomorphism” on A, leaves r fixed. This notion will be generalized to a relative one for any field extension L/K, and several results describing the set of f.a.f. elements are obtained.

Non-split nonassociative quaternion algebras over fields were first discovered over the real numbers independently by Dickson and Albert. They were later classified over arbitrary fields by Waterhouse. These algebras naturally appeared as the most interesting case in the classification of the four-dimensional nonassociative algebras which contain a separable field extension of the base field in their nucleus. We investigate algebras of constant rank 4 over an arbitrary ringR which contain a quadratic etale subalgebraS overR in their nucleus. A generalized Cayley-Dickson doubling process is introduced to construct a special class of these algebras.

Assume F is a homotopy invariant pseudo pretheory with torsion values and X is a smooth scheme of finite type over a field k. We show for certain field extensions k⊆K the map F(X)→F(XK) is an isomorphism.

All nondegenerate indecomposable forms of higher degree over a perfect field k can be realized as traces of nondegenerate absolutely indecomposable forms of higher degree over a suitable algebraic field extension of k. With the help of trace forms of certain nonassociative algebras we construct classes of indecomposable forms of degree d≥3.

Let K be a difference field with basic set of translations σ and let a partition of the set σ into p disjoint subsets be fixed. Let \(\mathcal{D}\) be the ring of difference operators over K treated as a filtered ring with the natural p-dimensional filtration. We introduce a special type of reduction in a finitely generated free \(\mathcal{D}\)-module and develop the corresponding technique of characteristic sets that allows one to prove that the Hilbert function in p variables associated with a finitely generated difference K-module is polynomial. We also outline a method of calculation of such a function and obtain a generalization of the author's theorem on difference dimension polynomials. In conclusion, we develop a similar technique of characteristic sets for the ring of difference polynomials and prove a theorem on multivariable difference dimension polynomials associated with a finitely generated difference field extension.

Let A p ⊂ C denote the set of all algebraic numbers such that α ∈ A p if and only if α is a zero of a (not necessarily irreducible) polynomial with positive rational coefficients. We give several results concerning the numbers in A p . In particular, the intersection of A p and the unit circle |z| = 1 is investigated in detail. So we determine all numbers of degree less than 6 on the unit circle which lie in the set A p . Further we show that when α is a root of an irreducible rational polynomial p(X) of degree ≠ 4 whose Galois group contains the full alternating group, α lies in A p if and only if no real root of p(X) is positive.

We previously described the Galois module structures of pth-power class groups K×/K×p, where K/F is a cyclic extension of degree p over a field F containing a primitive pth root of unity. That description relied upon arithmetic invariants associated with K/F. Here we construct field extensions K/F with prescribed arithmetic invariants, thus completing the classification of Galois modules K×/K×p.

Let F be a field, char F ≠ 2, and ϕ and ψ be anisotropic quadratic forms over F. Let L be a generic field extension of F such that i(ψ L ) ≥ 2. Under what conditions is the form ϕ L isotropic? We give an answer to this question in the cases where dim ϕ = 5, dim ψ = 6 and dim ϕ = 6, dim ψ = 7. Bibliography: 18 titles.

Recently Anderson described explicitly the epsilon extension of the maximal abelian ℚab of the rational number field ℚ, which is the compositum of all subfield of ℂ quadratic over ℚab and Galois over ℚ. We have given an analogous description of the epsilon extension of the maximal abelian extension of the rational function field over a finite field. In this paper we extend the theory of epsilon extensions partially to a global function field with a distinguished place. The new ingredient is that the base field may have non-trivial class group.

Let R be a complete discrete valuation ring with field of fractions K and let X K be a smooth, quasi-compact rigid-analytic space over Sp K. We show that there exists a finite separable field extension K' of K, a rigid-analytic space X' K' over Sp K' having a strictly semi-stable formal model over the ring of integers of K', and an etale, surjective morphism f : X' K' →X K of rigid-analytic spaces over Sp K. This is different from the alteration result of A.J. de Jong [dJ] who does not obtain that f is etale. To achieve this property we have to work locally on X K , i.e. our f is not proper and hence not an alteration.

Let R be a finite dimensional algebra over a field k. It is shown that a finite separable field extension of k preserves and respects generic tameness of R. Moreover if k is an infinite perfect field then the extension of k to its algebraic closure preserves generic tameness. As a corollary we get that if R is generically tame then for every number m all but a finite number of indecomposable R-modules of length m are DTr-periodic.

For some infinite algebraic extensionsM of the field of rational numbers we show that the integral closure of ℤ is first-order definable. As an application we prove that the Pythagorean hull of ℚ is an undecidable field.

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 $A_M$ .

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.

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.

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.