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A fundamental result of Springer says that a quadratic form over a field of characteristic ≠ 2 is isotropic if it is so after an odd degree field extension. In this paper we generalize Springer’s theorem as follows. Let R be an arbitrary semilocal ring, let S be a finite R -algebra of constant odd degree, which is etale or generated by one element, and let q be a nonsingular R -quadratic form whose base ring extension q S is isotropic. We show that then already q is isotropic.

We present a complete algorithm that computes all hypergeometric solutions of homogeneous linear difference equations and rational solutions of parameterized linear difference equations in the setting of Π Σ ⁎ -fields. More generally, we provide a flexible framework for a big class of difference fields that are built by a tower of Π Σ ⁎ -field extensions over a difference field that enjoys certain algorithmic properties. As a consequence one can compute all solutions in terms of indefinite nested sums and products that arise within the components of a parameterized linear difference equation, and one can find all hypergeometric solutions of a homogeneous linear difference equation that are defined over the arising sums and products.

Let K be a maximal unramified extension of a nonarchimedean local field with arbitrary residual characteristic p. Let G be a reductive group over K which splits over a tamely ramified extension of K. We show that the associated Moy-Prasad filtration representations are in a certain sense independent of p. We also establish descriptions of these representations in terms of explicit Weyl modules and as representations occurring in a generalized Vinberg-Levy theory. As an application, we use these results to provide necessary and sufficient conditions for the existence of stable vectors in Moy-Prasad filtration representations, which extend earlier results by Reeder and Yu (which required p to be large) and by Romano and the author (which required G to be absolutely simple and split). This yields new supercuspidal representations. We also treat reductive groups G that are not necessarily split over a tamely ramified field extension.

It is an open problem whether the Iwasawa λ-invariants of the Z p -extensions of a fixed number field are bounded. Using the class-field theoretic tool of logarithmic class groups, we obtain bounds for the λ-invariants of Z p -extensions of suitable field extensions of imaginary quadratic number fields. We also prove the Gross-Kuz'min Conjecture for certain families of non-cyclotomic Z p -extensions.

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.

This paper is a continuation of our previous works where we study maps from $X_0(N)$, $N \ge 1$, into $\mathbb P^2$ constructed via modular forms of the same weight and criteria that such a map is birational (see [12]). In the present paper our approach is based on the theory of primitive elements in finite separable field extensions. We prove that in most of the cases the constructed maps are birational, and we consider those such that the resulting equation of the image in $\mathbb P^2$ is simplest possible., Comment: arXiv admin note: text overlap with arXiv:1305.2428

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 determine the Hopf Galois structures on a Galois field extension of degree twice an odd prime square and classify the corresponding skew left braces. Besides we determine the separable field extensions of degree twice an odd prime square allowing a cyclic Hopf Galois structure and the number of these structures.

Given a collection A={L1,…,Ln} of intermediate fields in a field extension L/K of finite degree and ΛL,A=L1×⋯×Ln, there is a natural map ΨL,A:L→ΛL,A given by y↦(TrL/L1(y),…,TrL/Ln(y)), where TrL/Li...

We study the maximal multiplicity locus of a variety X over a field of characteristic p>0 that is provided with a finite surjective radicial morphism δ:X→V, where V is regular, for example, when X⊂An+1 is a hypersurface defined by an equation of the form Tq−f(x1,…,xn)=0 and δ is the projection onto V:=Spec(k[x1,…,xn]). The multiplicity along points of X is bounded by the degree, say d, of the field extension K(V)⊂K(X). We denote by Fd(X)⊂X the set of points of multiplicity d. Our guiding line is the search for invariants of singularities x∈Fd(X) with a good behavior property under blowups X′→X along regular centers included in Fd(X), which we call invariants with the pointwise inequality property. A finite radicial morphism δ:X→V as above will be expressed in terms of an OVq-submodule M⊆OV. A blowup X′→X along a regular equimultiple center included in Fd(X) induces a blowup V′→V along a regular center and a finite morphism δ′:X′→V′. A notion of transform of the OVq-module M⊂OV to an OV′q-module M′⊂OV′ will be defined in such a way that δ′:X′→V′ is the radicial morphism defined by M′. Our search for invariants relies on techniques involving differential operators on regular varieties and also on logarithmic differential operators. Indeed, the different invariants we introduce and the stratification they define will be expressed in terms of ideals obtained by evaluating differential operators of V on OVq-submodules M⊂OV.

This article investigates the cohomological kernels ker(H2n+1F→H2n+1E) of field extensions E/F in characteristic two where separable part of E is a quadratic extension F(α) and E is quadratic or qu...

Every Lie algebra over a field E gives rise to new Lie algebras over any subfield F ⊆ E by restricting the scalar multiplication. This paper studies the structure of these underlying Lie algebra in relation to the structure of the original Lie algebra, in particular the question how much of the original Lie algebra can be recovered from its underlying Lie algebra over subfields F. By introducing the conjugate of a Lie algebra we show that in some specific cases the Lie algebra is completely determined by its underlying Lie algebra. Furthermore we construct examples showing that these assumptions are necessary. As an application, we give for every positive n an example of a real 2-step nilpotent Lie algebra which has exactly n different bi-invariant complex structures. This answers an open question by Di Scala, Lauret and Vezzoni motivated by their work on quasi-Kahler Chern-flat manifolds in differential geometry. The methods we develop work for general Lie algebras and for general Galois extensions F ⊆ E , in contrast to the original question which only considered nilpotent Lie algebras of nilpotency class 2 and the field extension R ⊆ C . We demonstrate this increased generality by characterizing the complex Lie algebras of dimension ≤4 which are defined over R and over Q .

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.

We prove a bound on the number of primes with a given splitting behavior in a given field extension. This bound generalizes the Brun–Titchmarsh bound on the number of primes in an arithmetic progression. The proof is set up as an application of Selberg’s Sieve in number fields. The main new ingredient is an explicit counting result estimating the number of integral elements with certain properties up to multiplication by units. As a consequence of this result, we deduce an explicit estimate for the number of ideals of norm up to [Formula: see text].

Let $p$ be a prime, let $K$ be a complete discrete valuation field of characteristic $0$ with a perfect residue field of characteristic $p$, and let $G_{K}$ be the Galois group. Let $\unicode[STIX]{x1D70B}$ be a fixed uniformizer of $K$, let $K_{\infty }$ be the extension by adjoining to $K$ a system of compatible $p^{n}$th roots of $\unicode[STIX]{x1D70B}$ for all $n$, and let $L$ be the Galois closure of $K_{\infty }$. Using these field extensions, Caruso constructs the $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$-modules, which classify $p$-adic Galois representations of $G_{K}$. In this paper, we study locally analytic vectors in some period rings with respect to the $p$-adic Lie group $\operatorname{Gal}(L/K)$, in the spirit of the work by Berger and Colmez. Using these locally analytic vectors, and using the classical overconvergent $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules, we can establish the overconvergence property of the $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$-modules.

Let O be a differential graded (possibly colored) operad defined over rationals. Let us assume that there exists a zig-zag of quasi-isomorphisms connecting O ⊗ K to its cohomology , where K is any field extension of Q . We show that for a large class of such dg operads, a formality quasi-isomorphism for O exists and can be constructed recursively. Every step of our recursive procedure involves a solution of a finite dimensional linear system and it requires no explicit knowledge about the zig-zag of quasi-isomorphisms connecting O ⊗ K to its cohomology.

For any central simple algebra over a field F which contains a maximal subfield M with non-trivial automorphism group G = AutF(M), G is solvable if and only if the algebra contains a finite chain of subalgebras which are generalized cyclic algebras over their centers (field extensions of F) satisfying certain conditions. These subalgebras are related to a normal subseries of G. A crossed product algebra F is hence solvable if and only if it can be constructed out of such a finite chain of subalgebras. This result was stated for division crossed product algebras by Petit and overlaps with a similar result by Albert which, however, was not explicitly stated in these terms. In particular, every solvable crossed product division algebra is a generalized cyclic algebra over F.

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 present a method of Grobner bases with respect to several term orderings and use it to obtain new results on multivariate dimension polynomials of inversive difference modules. Then we use the difference structure of the module of Kahler differentials associated with a finitely generated inversive difference field extension of a given difference transcendence degree to describe the form of a multivariate difference dimension polynomial of the extension.

We study GL_2(F)-invariant periods on representations of GL_2(A), where F is a nonarchimedean local field and A/F a product of field extensions of total degree 3. For irreducible representations, a theorem of Prasad shows that the space of such periods has dimension at most 1, and is non-zero when a certain epsilon-factor condition holds. We give an extension of this result to a certain class of reducible representations (of Whittaker type), extending results of Harris--Scholl when A is the split algebra F x F x F., Comment: Revised version, to appear in Forum Mathematicum

Let $k$ be a field, $x_1, \dots, x_n$ be independent variables and $L_n = k(x_1, \dots, x_n)$. The symmetric group $��_n$ acts on $L_n$ by permuting the variables, and the projective linear group $\text{PGL}_2$ acts by \[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \colon x_i \mapsto \frac{a x_i + b}{c x_i + d} \] for each $i = 1, \ldots, n$. The fixed field $L_n^{\text{PGL}_2}$ is called "the field of cross-ratios". Given a subgroup $S \subset ��_n$, H. Tsunogai asked whether $L_n^S$ rational over $K_n^S$. When $n \geqslant 5$ the second author has shown that $L_n^S$ is rational over $K_n^S$ if and only if $S$ has an orbit of odd order in $\{ 1, \dots, n \}$. In this paper we answer Tsunogai's question for $n \leqslant 4$., 9 pages; to be appeared on the Canadian Mathematical Bulletin

We say that a field extension $L/F$ has the descent property for isometry (resp. similarity) of quadratic or symmetric bilinear forms if any two forms defined over $F$ that become isometric (resp. similar) over $L$ are already isometric (resp. similar) over $F$. The famous Artin-Springer theorem states that anisotropic quadratic or symmetric bilinear forms over a field stay anisotropic over an odd degree field extension. As a consequence, odd degree extensions have the descent property for isometry of quadratic as well as symmetric bilinear forms. While this is well known for nonsingular quadratic forms, it is perhaps less well known for arbitrary quadratic or symmetric bilinear forms in characteristic $2$. We provide a proof in this situation. More generally, we show that odd degree extensions also have the descent property for similarity. Moreover, for symmetric bilinear forms in characteristic $2$, one even has the descent property for isometry and for similarity for arbitrary separable algebraic extensions. We also show Scharlau's norm principle for arbitrary quadratic or bilinear forms in characteristic $2$., 17 pages

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.

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 prove the existence and determine some invariants of a Hilbert-type bivariate quasi-polynomial associated with a difference–differential field extension with weighted basic derivations and translations. We also show that such a quasi-polynomial can be expressed in terms of univariate Ehrhart quasi-polynomials of rational conic polytopes.

We investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions and one that computes Tamagawa numbers up to rational squares in general extensions. As an application, we extend some of the existing results on thep-parity conjecture for Selmer groups of abelian varieties by allowing more general local behaviour. We also give a complete classification of the behaviour of Tamagawa numbers for semistable 2-dimensional principally polarised abelian varieties that is similar to the well-known one for elliptic curves. The appendix explains how to use this classification for Jacobians of genus 2 hyperelliptic curves given by equations of the formy2=f(x), under some simplifying hypotheses.

We show that the $\ell$-adic Tate conjecture for divisors on smooth proper varieties over finitely generated fields of positive characteristic follows from the $\ell$-adic Tate conjecture for divisors on smooth projective surfaces over finite fields. Similar results for cycles of higher co-dimension are given., v1:6 pages. Comments are welcome. v2: Corrected a gap in the proof of Theorem 1.1.2. Changed Proposition 3.1.2 accordingly. v3: final version

We generalize a result of Orlov and Van den Bergh on the representability of a cohomological functor from the bounded derived category of a smooth projective variety over a field to the category of L-modules, to the case where L is a field extension of the base field k of the variety X, with L of transcendence degree less than or equal to one or L purely transcendental of degree 2. This result can be applied to investigate the behavior of an exact functor between the bounded derived categories of coherent sheaves of X and Y, with X and Y smooth projective and Y of dimension less than or equal to one or Y a rational surface. We show that for any such F there exists a "generic kernel" A in the derived category of the product, such that F is isomorphic to the Fourier-Mukai transform with kernel A after composing both with the pullback to the generic point of Y., Comment: to appear in Journal of Noncommutative Geometry

To generalize the notion of Galois closure for separable field extensions, we devise a notion of $G$-closure for algebras of commutative rings $R\to A$, where $A$ is locally free of rank $n$ as an $R$-module and $G$ is a subgroup of $\mathrm{S}_n$. A $G$-closure datum for $A$ over $R$ is an $R$-algebra homomorphism $\varphi: (A^{\otimes n})^{G}\to R$ satisfying certain properties, and we associate to a closure datum $\varphi$ a closure algebra $A^{\otimes n}\otimes_{(A^{\otimes n})^G} R$. This construction reproduces the normal closure of a finite separable field extension if $G$ is the corresponding Galois group. We describe G-closure data and algebras of finite \'etale algebras over a general connected ring $R$ in terms of the corresponding finite sets with continuous actions by the \'etale fundamental group of $R$. We show that if $2$ is invertible, then $\mathrm{A}_n$-closure data for free extensions correspond to square roots of the discriminant, and that $\mathrm{D}_4$-closure data for quartic monogenic extensions correspond to roots of the cubic resolvent., Comment: This is a condensed, updated, and revised version of the author's Ph.D.\ thesis

Let k be a perfect field such that for every n there are only finitely many field extensions, up to isomorphism, of k of degree n. If G is a reductive algebraic group defined over k, whose characteristic is very good for G, then we prove that G ( k ) has only finitely many z-classes. For each perfect field k which does not have the above finiteness property we show that there exist groups G over k such that G ( k ) has infinitely many z-classes.

We say that a smooth algebraic group $G$ over a field $k$ is very special if for any field extension $K/k$, every $G_K$-homogeneous $K$-variety has a $K$-rational point. It is known that every split solvable linear algebraic group is very special. In this note, we show that the converse holds, and discuss its relationship with the birational classification of algebraic group actions., Comment: Proof of Lemma 6 simplified following a suggestion of the referee

The present chapter is devoted to the basic theory of finite fields, including existence and uniqueness theorems as well as the main structural results. For this purpose, we also extend the fundamental material covered in Chapters 1 and 2 by proving several results on field extensions in general (in particular, in the first two sections).

In this note we show that any supersingular abelian variety is isogenous to a superspecial abelian variety without increasing field extensions. The proof uses minimal isogenies and the Galois descent. We then construct a superspecial abelian variety which not directly defined over a finite field. This answers negatively to a question of the author [J. Pure Appl. Alg., 2013] concerning of endomorphism algebras occurring in Shimura curves. Endomorphism algebras of supersingular elliptic curves over an arbitrary field are also investigated. We correct a main result of the author's paper [Math. Res. Let., 2010]., 14 pages. Revise Introduction; add Section 2; erratum to [Math. Res. Let., 2010]= arXiv:0905.0019

Let $A$ be a finite-dimensional $k$-algebra and $K/k$ be a finite separable field extension. We prove that $A$ is derived equivalent to a hereditary algebra if and only if so is $A\otimes_kK$., Comment: 7 pages

Let $A$ be a finite-dimensional algebra over a field $k$. We define $A$ to be $\mathbf{C}$-dichotomic if it has the dichotomy property of the representation type on complexes of projective $A$-modules. $\mathbf{C}$-dichotomy implies the dichotomy properties of representation type on the levels of homotopy category and derived category. If $k$ admits a finite separable field extension $K/k$ such that $K$ is algebraically closed, the real number field for example, we prove that $A$ is $\mathbf{C}$-dichotomic. As a consequence, the second derived Brauer-Thrall type theorem holds for $A$, i.e., $A$ is either derived discrete or strongly derived unbounded., Comment: 8 pages. Comments welcome!

We consider Hopf Galois structures on a separable field extension $L/K$ of degree $p^n$, for $p$ an odd prime number, $n\geq 3$. For $p > n$, we prove that $L/K$ has at most one abelian type of Hopf Galois structures. For a nonabelian group $N$ of order $p^n$, with commutator subgroup of order $p$, we prove that if $L/K$ has a Hopf Galois structure of type $N$, then it has a Hopf Galois structure of type $A$, where $A$ is an abelian group of order $p^n$ and having the same number of elements of order $p^m$ as $N$, for $1\leq m \leq n$., arXiv admin note: text overlap with arXiv:2003.01819

Let K be an inversive difference-differential field and L a (not necessarily inversive) finitely generated difference-differential field extension of K. We consider the natural filtration of the extension L/K associated with a finite system \(\eta \) of its difference-differential generators and prove that for any intermediate difference-differential field F, the transcendence degrees of the components of the induced filtration of F are expressed by a certain numerical polynomial \(\chi _{K, F,\eta }(t)\). This polynomial is closely connected with the dimension Hilbert-type polynomial of a submodule of the module of Kahler differentials \(\varOmega _{L^{*}|K}\) where \(L^{*}\) is the inversive closure of L. We prove some properties of polynomials \(\chi _{K, F,\eta }(t)\) and use them for the study of the Krull-type dimension of the extension L/K. In the last part of the paper, we present a generalization of the obtained results to multidimensional filtrations of L/K associated with partitions of the sets of basic derivations and translations.

Quantization is studied from a viewpoint of field extension. If the dynamical fields and their action have a periodicity, the space of wave functions should be algebraically extended `a la Galois, so that it may be consistent with the periodicity. This was pointed out by Y. Nambu three decades ago. Having chosen quantum mechanics (one dimensional field theory), this paper shows that a different Galois extension gives a different quantization scheme. A new scheme of quantization appears when the invariance under Galois group is imposed as a physical state condition. Then, the normalization condition appears as a sum over the product of more than three wave functions, each of which is given for a different root adjoined by the field extension.

We discuss isomorphism questions concerning the Hopf algebras that yield Hopf–Galois structures for a fixed separable field extension L/K. We study in detail the case where L/K is Galois with dihedral group \(D_p\), \(p\ge 3\) prime and give explicit descriptions of the Hopf algebras which act on L/K. We also determine when two such Hopf algebras are isomorphic, either as Hopf algebras or as algebras. For the case \(p=3\) and a chosen L/K, we give the Wedderburn–Artin decompositions of the Hopf algebras.

Let A ⊂ B be an integral ring extension of integral domains with fields of fractions K and L, respectively. The integral degree of A ⊂ B, denoted by dA(B), is defined as the supremum of the degrees of minimal integral equations of elements of B over A. It is an invariant that lies in between dK(L) and μA(B), the minimal number of generators of the A-module B. Our purpose is to study this invariant. We prove that it is sub-multiplicative and upper-semicontinuous in the following three cases: if A ⊂ B is simple; if A ⊂ B is projective and finite and K ⊂ L is a simple algebraic field extension; or if A is integrally closed. Furthermore, d is upper-semicontinuous if A is noetherian of dimension 1 and with finite integral closure. In general, however, d is neither sub-multiplicative nor upper-semicontinuous.

We introduce a notion of amenable normal extension S of a unital ring R with a finite approximation system F, encompassing the amenable algebras over a field of Gromov and Elek, the twisted crossed product by an amenable group, and the tensor product with a field extension. It is shown that every Sylvester matrix rank function rk of R preserved by S has a canonical extension to a Sylvester matrix rank function rk_F for S. In the case of twisted crossed product by an amenable group, and the tensor product with a field extension, it is also shown that rk_F depends on rk continuously. When an amenable group has a twisted action on a unital C*-algebra preserving a tracial state, we also show that two natural Sylvester matrix rank functions on the algebraic twisted crossed product constructed out of the tracial state coincide., Comment: Proposition 9.6 is added. 45 pages. To appear in J. Funct. Anal

In this paper, we study positive as well as non-negative and non-trivial gradings on finite dimensional Lie algebras. We give a different proof that the existence of such a grading on a Lie algebra is invariant under taking field extensions, a result very recently obtained by Y. Cornulier. Similarly, we prove that given a grading of one these types and a finite group of automorphisms, there always exist a positive grading which is preserved by this group. From these results we conclude that the existence of an expanding map or a non-trivial self-cover on an infra-nilmanifold depends only on the covering Lie group. Another application is the construction of a nilmanifold admitting an Anosov diffeomorphism but no non-trivial self-covers and in particular no expanding maps, which is the first known example of this type., Comment: 11 pages

We classify the split simple affine algebraic groups [math] of types A and C over a field with the property that the Chow group of the quotient variety [math] is torsion-free, where [math] is a special parabolic subgroup (e.g., a Borel subgroup) and [math] is a generic [math] -torsor (over a field extension of the base field). Examples of [math] include the adjoint groups of type A. Examples of [math] include the Severi–Brauer varieties of generic central simple algebras.

Let T/A be an integral extension of noetherian integrally closed integral domains whose quotient field extension is a finite cyclic Galois extension. Let S/R be a localization of this extension which is unramified. Using a generalized cyclic crossed product construction it is shown that certain reflexive fractional ideals of T with trivial norm give rise to Azumaya R-algebras that are split by S. Sufficient conditions on T/A are derived under which this construction can be reversed and the relative Brauer group of S/R is shown to fit into the exact sequence of Galois cohomology associated to the ramified covering T/A. Many examples of affine algebraic varieties are exhibited for which all of the computations are carried out.

A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a classification of finite-dimensional graded-central graded-division algebras over an arbitrary field [Formula: see text] can be reduced to the following three classifications, for each finite Galois extension [Formula: see text] of [Formula: see text]: (1) finite-dimensional central division algebras over [Formula: see text], up to isomorphism; (2) twisted group algebras of finite groups over [Formula: see text], up to graded-isomorphism; (3) [Formula: see text]-forms of certain graded matrix algebras with coefficients in [Formula: see text] where [Formula: see text] is as in (1) and [Formula: see text] is as in (2). As an application, we classify, up to graded-isomorphism, the finite-dimensional graded-division algebras over the field of real numbers (or any real closed field) with an abelian grading group. We also discuss group gradings on fields.