Let D be a division ring. We say that D is left algebraic over a (not necessarily central) subfield K of D if every x in D satisfies a polynomial equation x^n + a_{n-1}x^{n-1}+...+a_0=0 with a_0,...,a_{n-1} in K. We show that if D is a division ring that is left algebraic over a subfield K of bounded degree d then D is at most d^2-dimensional over its center. This generalizes a result of Kaplansky. For the proof we give a new version of the combinatorial theorem of Shirshov that sufficiently long words over a finite alphabet contain either a q-decomposable subword or a high power of a non-trivial subword. We show that if the word does not contain high powers then the factors in the q-decomposition may be chosen to be of almost the same length. We conclude by giving a list of problems for algebras that are left algebraic over a commutative subring.
We show in this article that in many cases the subfields of a nondegenerate tame semiramified division algebra of prime power degree over a Henselian valued field are inertial field extensions of the center.
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 $D$ be a division algebra over a field $F$. In this paper, we prove that there exist $a,b,x,y\in D^*$ such that $F(ab-ba)$ and $F(xyx^{-1}y^{-1})$ are maximal subfields of $D$, which answers questions posted in [5].
We address the problem of when two finite dimensional central division algebras over the same field are necessarily isomorphic given that they have the same maximal subfields.
Let D be a valued division algebra, finite-dimensional over its center F. Assume D has an unramified splitting field. The paper shows that if D contains a maximal subfield which is Galois over F (i.e. D is a crossed product) then the residue division algebra of D contains a maximal subfield which is Galois over the residue field of F. This theorem captures an essential argument of previously known noncrossed product proofs in the more general language of noncommutative valuations. The result is particularly useful in connection with explicit constructions.
Let D be a division ring finite dimensional over its center F. The goal of this paper is to prove that for any positive integer n there exists a in D^(n); the n-th multiplicative derived subgroup, such that F(a) is a maximal subfield of D. We also show that a single depth-n iterated additive commutator would generate a maximal subfield of D.
Let D be a division algebra with center F. A maximal subfield of D is defined to be a field K such that CD(K) = K; that is, K is its own centralizer in D. A maximal subfield K is said to be self-invariant if it normalises by itself, i.e. ND*(K)= K: This kind of subfields is important because they have strong connexion with the most famous Albert's Conjecture (every division ring of prime index is cyclic). In fact, we pose a question that asserts whether every division ring whose all maximal subfields are self-invariant has to be commutative. The positive answer to this question, in finite dime…
For any number field K, it is unknown which finite groups appear as Galois groups of extensions L/K such that L is a maximal subfield of a division algebra with center K (a K-division algebra). For K=Q, the answer is described by the long standing Q-admissibility conjecture. We extend a theorem of Neukirch on embedding problems with local constraints in order to determine for every number field K, what finite solvable groups G appear as Galois groups of tame maximal subfields of K-division algebras, generalizing Liedahl's theorem for metacyclic G and Sonn's solution of the Q-admissibility conj…
By generalizing the method used by Tignol and Amitsur in [TA85], we determine necessary and sufficient conditions for an arbitrary tame central division algebra D over a Henselian valued field E to have Kummer subfields [Corollary 2.11 and Corollary 2.12]. We prove also that if D is a tame semiramified division algebra of prime power degree p^n over E such that p\neq char(\bar E) and rk(Γ_D/Γ_E)\geq 3 [resp., such that p\neq char(\bar E) and p^3 divides exp(Γ_D/Γ_E)], then D is non-cyclic [Proposition 3.1] [resp., D is not an elementary abelian crossed product [Proposition 3.2]].
We give a survey of recent results related to the problem of characterizing finite-dimensional division algebras by the set of isomorphism classes of their maximal subfields. We also discuss various generalizations of this problem and some of its applications. In the last section, we extend the problem to the context of absolutely almost simple algebraic groups.
We describe relations between maximal subfields in a division ring and in its rational extensions. More precisely, we prove that properties such as being Galois or purely inseparable over the centre generically carry over from one to another. We provide an application to enveloping skewfields in positive characteristics. Namely, there always exist two maximal subfields of the enveloping skewfield of a solvable Lie algebra, such that one is Galois and the second purely inseparable of exponent 1 over the centre. This extends results of Schue in the restricted case. Along the way we provide a des…
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.
The genus $gen(D)$ of a finite-dimensional central division algebra $D$ over a field $F$ is defined as the collection of classes $[D']\in Br(F)$, where $D'$ is a central division $F$-algebra having the same maximal subfields as $D$. We show that the fact that quaternion division algebras $D$ and $D'$ have the same maximal subfields does not imply that the matrix algebras $M_l(D)$ and $M_l(D')$ have the same maximal subfields for $l>1$. Moreover, for any odd $n>1$, we construct a field $L$ such that there are two quaternion division $L$-algebras $D$ and $D'$ and a central division $L$-algebra $…
A celebrated theorem of P.M.Cohn says that for any two division rings (not necessarily finite dimensional) over a field F, their amalgamated product over F is a domain which can be embedded in a division ring. Note that even with the two initial division rings begin finite dimensional over their centers, the resulting division ring is never finite dimensional over its center. Perhaps this led Lance Small to ask the following question. Assume $F_1$ and $F_2$ are fields with the same characteristic. Small asked whether any two division algebras $D_1/F_1$ and $D_2/F_2$ can be embedded in some thi…
We investigate the finite subgroups that occur in the Hamiltonian quaternion algebra over the real subfield of cyclotomic fields. When possible, we investigate their distribution among the maximal orders.
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,…
Let $D$ be a division ring with the center $F$ and $D^*$ be the multiplicative group of $D$. In this paper we study locally nilpotent maximal subgroups of $D^*$. We give some conditions that influence the existence of locally nilpotent maximal subgroups in division ring with infinite center. Also, it is shown that if $M$ is a locally nilpotent maximal subgroup that is algebraic over $F$, then either it is the multiplicative group of some maximal subfield of $D$ or it is center by locally finite. If, in addition we assume that $F$ is finite and $M$ is nilpotent, then the second case cannot occu…
We show that if two division $p$-algebras of prime degree share an inseparable field extension of the center then they also share a cyclic separable one. We show that the converse is in general not true. We also point out that sharing all the inseparable field extensions of the center does not imply sharing all the cyclic separable ones.
A finite group G is admissible over a field M if there is a division algebra whose center is M with a maximal subfield G-Galois over M. We consider nine possible notions of being admissible over M with respect to a subfield K of M, where the division algebra, the maximal subfield or the Galois group are asserted to be defined over K. We completely determine the logical implications between all variants.
We study the existence of free subalgebras in division algebras, and prove the following general result: if $A$ is a noetherian domain which is countably generated over an uncountable algebraically closed field $k$ of characteristic 0, then either the quotient division algebra of $A$ contains a free algebra on two generators, or it is left algebraic over every maximal subfield. As an application, we prove that if $k$ is an uncountable algebraically closed field and $A$ is a finitely generated $k$-algebra that is a domain of GK-dimension strictly less than 3, then either $A$ satisfies a polynom…
Let $\mathbb{D}$ be a division ring and $\mathbb{F}$ be a subfield of the center of $\mathbb{D}$ over which $\mathbb{D}$ has finite dimension $d$. Let $n,p,r$ be positive integers and $\mathcal{V}$ be an affine subspace of the $\mathbb{F}$-vector space $M_{n,p}(\mathbb{D})$ in which every matrix has rank less than or equal to $r$. Using a new method, we prove that $\dim_{\mathbb{F}} \mathcal{V} \leq \max(n,p)\,rd$ and we characterize the spaces for which equality holds. This extends a famous theorem of Flanders which was known only for fields.
A finite group G is called admissible over a given field if there exists a central division algebra that contains a G-Galois field extension as a maximal subfield. We give a definition of embedding problems of division algebras that extends both the notion of embedding problems of fields as in classical Galois theory, and the question which finite groups are admissible over a field. In a recent work by Harbater, Hartmann and Krashen, all admissible groups over function fields of curves over complete discretely valued fields with algebraically closed residue field of characteristic zero have be…
Let D be a central division algebra of degree n over a field K. One defines the genus gen(D) of D as the set of classes [D'] in the Brauer group Br(K) of K represented by central division algebras D' of degree n over K having the same maximal subfields as D. We prove that if the field K is finitely generated and n is prime to its characteristic, then gen(D) is finite, and give explicit estimations of its size in certain situations.
We propose polynomial-time algorithms for finding nontrivial zeros of quadratic forms with four variables over rational function fields of characteristic 2. We apply these results to find prescribed quadratic subfields of quaternion division division algebras and zero divisors in $M_2(D)$, the full matrix algebra over a division algebra, given by structure constants. We also provide an implementation of our results in MAGMA which shows that the algorithms are truly practical.
Let $D$ be a division ring and $K$ a subfield of $D$ which is not necessarily contained in the center $F$ of $D$. In this paper, we study the structure of $D$ under the condition of left algebraicity of certain subsets of $D$ over $K$. Among results, it is proved that if $D^*$ contains a noncentral normal subgroup which is left algebraic over $K$ of bounded degree $d$, then $[D:F]\le d^2$. In case $K=F$, the obtained results show that if either all additive commutators or all multiplicative commutators with respect to a noncentral subnormal subgroup of $D^*$ are algebraic of bounded degree $d$…
Let $k$ be an uncountable algebraically closed field and let $A$ be a countably generated left Noetherian $k$-algebra. Then we show that $A \otimes_k K$ is left Noetherian for any field extension $K$ of $k$. We conclude that all subfields of the quotient division algebra of a countably generated left Noetherian domain over $k$ are finitely generated extensions of $k$. We give examples which show that $A\otimes_k K$ need not remain left Noetherian if the hypotheses are weakened.
Let $A$ be a finite-dimensional division algebra containing a base field $k$ in its center $F$. We say that $A$ is defined over a subfield $F_0$ of $F$ if $A = A_0\otimes_{F_0} F$ for some $F_0$-subalgebra $A_0$ of $A$. We show that: (1) In many cases $A$ can be defined over a rational extension of $k$. (2) If $A$ has odd degree $n \ge 5$, then $A$ is defined over a field $F_0$ of transcendence degree at most $(n-1)(n-2)/2$ over $k$. (3) If $A$ is a $Z/m \times Z/2$-crossed product for some $m \ge 2$ (and in particular, if $A$ is any algebra of degree 4) then $A$ is Brauer equivalent to a tens…
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…
