Maximal Subfield Research Articles

Admissibility over semi-global fields in the bad characteristic case

A finite group G is said to be admissible over a field F if there exists a division algebra D central over F with a maximal subfield L such that L/F is Galois with group G. In this paper we give a complete characterization of admissible groups over function fields of curves over equicharacteristic complete discretely valued fields with algebraically closed residue fields, such as the field FP‾((t))(x).

Splitting fields of differential symbol algebras

For m≥2, we study derivations on symbol algebras of degree m over fields with characteristic not dividing m. A differential central simple algebra over a field k is split by a finitely generated extension of k. For certain derivations on symbol algebras, we provide explicit construction of differential splitting fields and give bounds on their algebraic and transcendence degrees. We further analyze maximal subfields that split certain differential symbol algebras.

Linkage of sets of cyclic algebras

Let p be a prime integer and F the function field in two algebraically independent variables over a smaller field F 0 . We prove that if char ( F 0 ) = p ⩾ 3 then there exist p 2 − 1 cyclic algebras of degree p over F that have no maximal subfield in common, and if char ( F 0 ) = 0 then there exist p 2 cyclic algebras of degree p over F that have no maximal subfield in common.

Subnormal subgroups and self-invariant maximal subfields in division rings

A maximal subfield of a division ring is said to be self-invariant if it is its own normalizer. Subfields of this kind are important because they have a strong connection with Albert's conjecture on the cyclicity of division rings of prime index. We show that every maximal subfield of the Mal'cev-Neumann division ring, which is of infinite dimension, is self-invariant. We also apply the Mal'cev-Neumann structure to refute the conjecture that every noncentral subnormal subgroup of the multiplicative group of a division ring must contain a noncentral normal subgroup. Finally, among other things, we rely on self-invariant subfields to present a criterion for a division ring to have a finite-dimensional subdivision rings.

Computation of residual polynomial operators of inductive valuations

Let ( K , v ) be a valued field, and μ an inductive valuation on K [ x ] extending v . Let G μ be the graded algebra of μ over K [ x ] , and κ the maximal subfield of the subring of G μ formed by the homogeneous elements of degree zero. In this paper, we find an algorithm to compute the field κ and the residual polynomial operator R μ : K [ x ] → κ [ y ] , where y is another indeterminate, without any need to perform computations in the graded algebra. This leads to an OM algorithm to compute the factorization of separable defectless polynomials over henselian fields.

On Crossed Product Algebras over Henselian Valued Fields

Let D be a tame central division algebra over a Henselian valued field E, [Formula: see text] be the residue division algebra of D, [Formula: see text] be the residue field of E, and n be a positive integer. We prove that Mn([Formula: see text]) has a strictly maximal subfield which is Galois (resp., abelian) over [Formula: see text] if and only if Mn(D) has a strictly maximal subfield K which is Galois (resp., abelian) and tame over E with ΓK ⊆ ΓD, where ΓK and ΓD are the value groups of K and D, respectively. This partially generalizes the result proved by Hanke et al. in 2016 for the case n = 1.

Diophantine Equation Generated by the Maximal Subfield of a Circular Field

Using the fundamental basis of the field $L_9=\mathbb{Q} (2\cos(\pi/9)),$ the form $N_{L_9}(\gamma)=f(x, y, z)$ is found and the Diophantine equation $f(x,y,z)=a$ is solved. A similar scheme is used to construct the form $N_{L_7}(\gamma)=g(x,y,z)$ . The Diophantine equation $g (x, y, z)=a$ is solved.

On genus of division algebras

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 $C$ of degree and exponent $n$ such that $gen(D) = gen(D')$ but $gen(D \otimes C) \ne gen(D' \otimes C)$.

Minimal Proper Quasifields with Additional Conditions

We investigate the finite semifields which are distributive quasifields, and finite near-fields which are associative quasifields. A quasifield Q is said to be a minimal proper quasifield if any of its sub-quasifield H ̸= Q is a subfield. It turns out that there exists a minimal proper near-field such that its multiplicative group is a Miller–Moreno group. We obtain an algorithm for constructing a minimal proper near-field with the number of maximal subfields greater than fixed natural number. Thus, we find the answer to the question: Does there exist an integer N such that the number of maximal subfields in arbitrary finite near-field is less than N? We prove that any semifield of order p4 (p be prime) is a minimal proper semifield

Диофантово уравнение, порожденное максимальным подполем кругового поля

Диофантово уравнение, порожденное максимальным подполем кругового поля

Cohomological kernels of non-normal extensions in characteristic two and indecomposable division algebras of index eight

This article investigates the cohomological kernels of field extensions E/F in characteristic two where separable part of E is a quadratic extension and E is quadratic or quartic and purely inseparable over Explicit generators for these kernels are described in both the normal and non-normal cases. The second part of the article gives a construction of an indecomposable division algebra D of index 8 and exponent 2 in characteristic two. The non-normal extensions E/F just considered can arise as maximal subfields of D while in the normal case such a maximal subfield will force a division algebra of exponent 2 to decompose.

English

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 ∈ D(n), the nth 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.

SOLVABLE CROSSED PRODUCT ALGEBRAS REVISITED

AbstractFor 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.

On some 3-primitive projective planes

We evolve an approach to construction and classification of semifield projective planes with the use of the linear space and spread set. This approach is applied to the problem of existance for a projective plane with the fixed restrictions on collineation group. A projective plane is said to be semifield plane if its coordinatizing set is a semifield, or division ring. It is an algebraic structure with two binary operation which satisfies all the axioms for a skewfield except (possibly) associativity of multiplication. A collineation of a projective plane of order p 2n ( p > 2 be prime) is called Baer collineation if it fixes a subplane of order p n pointwise. If the order of a Baer collineation divides p n − 1 but does not divide p i − 1 for i < n then such a collineation is called p -primitive. A semifield plane that admit such collineation is a p -primitive plane. M. Cordero in 1997 construct 4 examples of 3-primitive semifield planes of order 81 with the nucleus of order 9, using a spread set formed by 2 × 2-matrices. In the paper we consider the general case of 3-primitive semifield plane of order 81 with the nucleus of order ≤ 9 and a spread set in the ring of 4 × 4-matrices. We use the earlier theoretical results obtained independently to construct the matrix representation of the spread set and autotopism group. We determine 8 isomorphism classes of 3-primitive semifield planes of order 81 including M. Cordero examples. We obtain the algorithm to optimize the identification of pair-isomorphic semifield planes, and computer realization of this algorithm. It is proved that full collineation group of any semifield plane of order 81 is solvable, the orders of all autotopisms are calculated. We describe the structure of 8 non-isotopic semifields of order 81 that coordinatize 8 nonisomorphic 3-primitive semifield planes of order 81. The spectra of its multiplicative loops of non-zero elements are calculated, the left-, right-ordered spectra, the maximal subfields and automorphisms are found. The results obtained illustrate G. Wene hypothesis on left or right primitivity for any finite semifield and demonstrate some anomalous properties. The methods and algorithsm demonstrated can be used for construction and investigation of semifield planes of odd order p n for p ≥ 3 and n ≥ 4.

Counting and effective rigidity in algebra and geometry

The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length that can be explicitly bounded as a function of volume. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3-manifolds. These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest.

Triple linkage of quadratic Pfister forms

Given a field F of characteristic 2, we prove that if every three quadratic n-fold Pfister forms have a common quadratic $$(n-1)$$ -fold Pfister factor then $$I_q^{n+1} F=0$$ . As a result, we obtain that if every three quaternion algebras over F share a common maximal subfield then u(F) is either 0, 2 or 4. We also prove that if F is a nonreal field with $${\text {char}}(F) \ne ~2$$ and $$u(F)=4$$ , then every three quaternion algebras share a common maximal subfield.

Involutions and stable subalgebras

Given a central simple algebra with involution over an arbitrary field, étale subalgebras contained in the space of symmetric elements are investigated. The method emphasizes the similarities between the various types of involutions and privileges a unified treatment for all characteristics whenever possible. As a consequence a conceptual proof of a theorem of Rowen is obtained, which asserts that every division algebra of exponent two and degree eight contains a maximal subfield that is a triquadratic extension of the centre.

Problems on structure of finite quasifields and projective translation planes

It is well-known that the constructions and classification of non-Desarguesian projective planes are closely connected with ones for quasifields. We consider the problems on structure of finite quasifields and semifields: automorphisms and autotopisms, maximal subfields and their orders, the spectrum of orders of non-zero elements and hypotheses about generated subsets of the multiplicative loop.

Regulators of an Infinite Family of the Simplest Quartic Function Fields

AbstractWe explicitly find regulators of an infinite family {Lm} of the simplest quartic function fields with a parameter m in a polynomial ring [t], where is the finite field of order q with odd characteristic. In fact, this infinite family of the simplest quartic function fields are subfields of maximal real subfields of cyclotomic function fields having the same conductors. We obtain a lower bound on the class numbers of the family {Lm} and some result on the divisibility of the divisor class numbers of cyclotomic function fields that contain {Lm} as their subfields. Furthermore, we find an explicit criterion for the characterization of splitting types of all the primes of the rational function field (t) in {Lm}.

An application of subgroup lattices

We give a lattice theoretic proof of the well-known result that a finite group G is cyclic iff G has at most one subgroup of each order dividing |G|. Consequently, we show that a division ring D is a field iff D has at most one maximal subfield.