Let $p$ be a prime number and $(K, v)$ a Henselian valued field with a residue field $\widehat K$. This paper determines the Brauer $p$-dimension of $K$, in case $p \neq {\rm char}(\widehat K)$ and $\widehat K$ is a $p$-quasilocal field properly included in its maximal $p$-extension. When $\widehat K$ is a local field, it describes index-exponent pairs of central division $K$-algebras of $p$-primary degrees. The same goal is achieved, if $(K, v)$ is maximally complete, char$(K) = p$ and $\widehat K$ is local.
Let $E$ be a primarily quasilocal field, $M/E$ a finite Galois extension and $D$ a central division $E$-algebra of index divisible by $[M\colon E]$. In addition to the main result of Part I, this part of the paper shows that if the Galois group $G(M/E)$ is not nilpotent, then $M$ does not necessarily embed in $D$ as an $E$-subalgebra. When $E$ is quasilocal, we find the structure of the character group of its absolute Galois group; this enables us to prove that if $E$ is strictly quasilocal and almost perfect, then the divisible part of the multiplicative group $E ^{\ast}$ equals the intersect…
Let $F$ be a Henselian field of $q$-cohomological dimension $3$, where $q$ is a prime. Let $Γ_F$ be the totally ordered abelian value group of $F$ and let $D$ be a central division algebra over $F$ of index a power of $q$ such that the characteristic of the residue field, $\overline{F}$ is coprime to $q$. We show that when $1\leq\dim_{\mathbb{F}_q}(Γ_F/qΓ_F)\leq 3$, the reduced Whitehead group of $D$ is trivial.
Let $(K, v)$ be a Henselian valued field satisfying the following conditions, for a given prime number $p$: (i) central division $K$-algebras of (finite) $p$-primary dimensions have Schur indices equal to their exponents; (ii) the value group $v(K)$ properly includes its subgroup $pv(K)$. The paper shows that if $\hat K$ is the residue field of $(K, v)$ and $\hat R$ is an intermediate field of the maximal $p$-extension $\hat K (p)/\hat K$, then the natural homomorphism Br$(\hat K) \to $ Br$(\hat R)$ of Brauer groups maps surjectively the $p$-component Br$(\hat K)_{p}$ on Br$(\hat R)_{p}$. It p…
The epicenter of this paper concerns Pfister quadratic forms over a field $F$ with a Henselian discrete valuation. All characteristics are considered but we focus on the most complicated case where the residue field has characteristic 2 but $F$ does not. We also prove results about round quadratic forms, composition algebras, generalizations of composition algebras we call conic algebras, and central simple associative symbol algebras. Finally we give relationships between these objects and Kato's filtration on the Milnor $K$-groups of $F$.
The existence of finite dimensional central division algebras with no maximal subfield that is Galois over the center (called noncrossed products), was for a time the biggest open problem in the theory of division algebras, before it was settled by Amitsur. Motivated by Brussel's discovery of noncrossed products over Q((t)), we describe the "location" of noncrossed products in the Brauer group of general Henselian valued fields with arbitrary value group and global residue field. We show that within the fibers defined canonically by Witt's decomposition of the Brauer group of such fields, cros…
Let $(K, v)$ be a Henselian discrete valued field with residue field $\widehat K$ of characteristic $p$, and Brd$_{p}(K)$ be the Brauer $p$-dimension of $K$. This paper shows that Brd$_{p}(K) \ge n$, if $[\widehat K\colon \widehat K ^{p}] = p ^{n}$, for some $n \in \mathbb{N}$. It proves that Brd$_{p}(K) = \infty $ if and only if $[\widehat K\colon \widehat K ^{p}] = \infty $.
This paper determines the Brauer $p$-dimension Brd$_{p}(K)$ and the absolute Brauer $p$-dimension abrd$_{p}(K)$ of a Henselian valued field $(K, v)$, for a prime $p \neq {\rm char}(\widehat K)$, under restrictions on the residue field $\widehat K$, such as the condition abrd$_{p}(\widehat K) = 0$. It describes the set $Σ_{0}$ of sequences ${\rm abrd}_{p}(E), {\rm Brd}_{p}(E)$, $p \in \mathbb P$, where $\mathbb P$ is the set of prime numbers and $E$ runs across the class of Henselian fields with char$(\widehat E) = 0$ and a projective absolute Galois group $\mathcal{G}_{\widehat E}$. Specifical…
We initiate the study of definable V-topolgies and show that there is at most one such V-topology on a t-henselian NIP field. Equivalently, we show that if $(K,v_1,v_2)$ is a bi-valued NIP field with $v_1$ henselian (resp. t-henselian) then $v_1$ and $v_2$ are comparable (resp. dependent). As a consequence Shelah's conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting a henselian valuation with a dp-minimal residue field. We conclude by showing that Shelah's conjecture is equivalent to the statement th…
We show that a finite-dimensional tame division algebra D over a Henselian field F has a maximal subfield Galois over F if and only if its residue division algebra has a maximal subfield Galois over the residue field of F. This generalizes the mechanism behind several known noncrossed product constructions to a crossed product criterion for all tame division algebras, and in particular for all division algebras if the residue characteristic is 0. If the residue field is a global field, the criterion leads to a description of the location of noncrossed products among tame division algebras, and…
Let $(K, v)$ be a Henselian discrete valued field with residue field $\widehat K$ of characteristic $q \ge 0$, and Brd$_{p}(K)$ be the Brauer $p$-dimension of $K$, for each prime $p$. The present paper shows that if $p = q$, then Brd$_{p}(K) \le 1$ if and only if $\widehat K$ is a $p$-quasilocal field and the degree $[\widehat K\colon \widehat K ^{p}]$ is $\le p$. This complements our earlier result that, in case $p \neq q$, we have Brd$_{p}(K) \le 1$ if and only if $\widehat K$ is $p$-quasilocal and Brd$_{p}(\widehat K) \le 1$.
Let $R$ be a commutative local ring. It is proved that $R$ is Henselian if and only if each $R$-algebra which is a direct limit of module finite $R$-algebras is strongly clean. So, the matrix ring $\mathbb{M}_n(R)$ is strongly clean for each integer $n>0$ if $R$ is Henselian and we show that the converse holds if either the residue class field of $R$ is algebraically closed or $R$ is an integrally closed domain or $R$ is a valuation ring. It is also shown that each $R$-algebra which is locally a direct limit of module-finite algebras, is strongly clean if $R$ is a $π$-regular commutative ring.
Let $K$ be the fraction field of a 2-dimensional, henselian, excellent local domain with finite residue field $k$. When the characteristic of $k$ is not 2, we prove that every quadratic form of rank $\ge 9$ is isotropic over $K$ using methods of Parimala and Suresh, and we obtain the local-global principle for isotropy of quadratic forms of rank 5 with respect to discrete valuations of $K$. The latter result is proved by making a careful study of ramification and cyclicity of division algebras over the field $K$, following Saltman's methods. A key step is the proof of the following result, whi…
Let A be a central simple algebra with involution sigma of first or second kind. Let v be a valuation on the sigma-fixed part F of Z(A). A sigma-special v-gauge g on A is a kind of value function on A extending v on F, such that g(sigma(x) x) = 2g(x) for all x in A. It is shown (under certain restrictions if the residue characteristic is 2) that if v is Henselian, then there is a sigma-special v-gauge g if and only if sigma is anisotropic, and g is unique. If v is not Henselian, it is shown that there is a sigma-special v-gauge g if and only if sigma remains anisotropic after scalar extension …
If K/F is a finite abelian Galois extension of global fields whose Galois group has exponent t, we prove that there exists a short exact sequence that has as a consequence that if t is square free, then Dec(K/F)=Br_{t}(K/F) which we use to show that prime exponent division algebras over Henselian valued fields with global residue fields are isomorphic to a tensor product of cyclic algebras. Finally, we construct a counterexample to the result for higher exponent division algebras.
We generalize the Hasse invariant of local class field theory to the tame Brauer group of a higher dimensional local field, and use it to study the arithmetic of central simple algebras over such fields, which are given {\it a priori} as tensor products of standard cyclic algebras. We also compute the tame Brauer dimension (or {\it period-index bound}) and the cyclic length of a general henselian-valued field of finite rank and finite residue field.
For a Noetherian local ring (R, m) having a finite residue field of cardinality q, we study the connections between the ideal Z(R) of R[x], which is the set of polynomials that vanish on R, and the ideal Z(m), the polynomials that vanish on m, using what we call pi-polynomials: polynomials of the form p(x) = \prod_{i = 1}^{q} (x - c_i), where c1, ..., cq is a set of representatives of the residue classes of m. When R is Henselian we prove that p(R) = m and show that a generating set for Z(R) may be obtained from a generating set for Z(m) by composing with p(x). When m is principal and has inde…
A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper provides an existence theorem for such fields over global fields with enough roots of unity. As an application, all twisted function fields and all twisted Laurent series rings over symbol algebras (or p-algebras) over global fields are crossed products. A closely related statement holds for division algebras over Henselian valued fields with global residue field. The existence of absolute Galois splitting fields in central simple algebras…
This paper proves that if $E$ is a field, such that the Galois group $\mathcal{G}(E(p)/E)$ of the maximal $p$-extension $E(p)/E$ is a Demushkin group of finite rank $r(p)_{E} \ge 3$, for some prime number $p$, then $\mathcal{G}(E(p)/E)$ does not possess nontrivial proper decomposition groups. When $r(p)_{E} = 2$, it describes the decomposition groups of $\mathcal{G}(E(p)/E)$. The paper shows that if $(K, v)$ is a $p$-Henselian valued field with $r(p)_{K} \in \mathbb N$ and a residue field of characteristic $p$, then $P \cong \widetilde P$ or $P$ is presentable as a semidirect product $\mathbb …
