Division Algebras of Prime Degree and Maximal Galois p-Extensions

2007 ◽  
Vol 59 (3) ◽  
pp. 658-672
Author(s):  
J. Mináč ◽  
A. Wadsworth
Keyword(s):  
Prime Number ◽  
Division Algebra ◽  
Roots Of Unity ◽  
Division Algebras ◽  
Prime Degree ◽  
Cyclic Algebra ◽  
Residue Fields ◽  
Residue Characteristic

AbstractLet p be an odd prime number, and let F be a field of characteristic not p and not containing the group μp of p-th roots of unity. We consider cyclic p-algebras over F by descent from L = F(μp). We generalize a theorem of Albert by showing that if μpn ⊆ L, then a division algebra D of degree pn over F is a cyclic algebra if and only if there is d ∈ D with dpn ∈ F – Fp. Let F(p) be the maximal p-extension of F. We show that F(p) has a noncyclic algebra of degree p if and only if a certain eigencomponent of the p-torsion of Br(F(p)(μp)) is nontrivial. To get a better understanding of F(p), we consider the valuations on F(p) with residue characteristic not p, and determine what residue fields and value groups can occur. Our results support the conjecture that the p torsion in Br(F(p)) is always trivial.

On the non-existence of cyclic splitting fields for division algebras

2018 ◽  
Vol 30 (2) ◽  
pp. 385-395
Author(s):  
Mehran Motiee
Keyword(s):  
Division Algebra ◽  
Commutator Subgroup ◽  
Cyclic Extension ◽  
Roots Of Unity ◽  
Division Algebras ◽  
Group Of Units ◽  
Positive Integers ◽  
Special Classes

AbstractLetDbe a division algebra over its centerFof degreen. Consider the group{\mu_{Z}(D)=\mu_{n}(F)/Z(D^{\prime})}, where{\mu_{n}(F)}is the group of all then-th roots of unity in{F^{*}}, and{Z(D^{\prime})}is the center of the commutator subgroup of the group of units{D^{*}}ofD. It is shown that if{\mu_{Z}(D\otimes_{F}L)\neq 1}for someLcontaining all the primitive{n^{k}}-th roots of unity for all positive integersk, thenDis not split by any cyclic extension ofF. This criterion is employed to prove that some special classes of division algebras are not cyclically split.

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

scholarly journals On the Automorphisms of the Four-dimensional Real Division Algebras

2017 ◽  
Vol 9 (2) ◽  
pp. 95
Author(s):  
Andre S. Diabang ◽  
Alassane Diouf ◽  
Mankagna A. Diompy ◽  
Alhousseynou Ba
Keyword(s):  
Division Algebra ◽  
Division Algebras ◽  
Automorphisms Groups

In this paper, we study partially the automorphisms groups of four-dimensional division algebra. We have proved that there is an equivalence between Der(A)=su(2) and Aut(A)=SO(3). For an unitary four-dimensional real division algebra, there is an equivalence between dim(Der(A))=1 and Aut(A)=SO(2).

A Note on Subnormal Subgroups of Division Algebras

1978 ◽  
Vol 30 (01) ◽  
pp. 161-163 ◽  
Author(s):  
Gary R. Greenfield
Keyword(s):  
Division Algebra ◽  
Multiplicative Group ◽  
Subnormal Subgroup ◽  
Local Fields ◽  
Division Algebras ◽  
Subnormal Subgroups

Let D be a division algebra and let D* denote the multiplicative group of nonzero elements of D. In [3] Herstein and Scott asked whether any subnormal subgroup of D* must be normal in D*. Our purpose here is to show that division algebras over certain p-local fields do not satisfy such a “subnormal property”.

scholarly journals Universal and homogeneous embeddings of dual polar spaces of rank 3 defined over quadratic alternative division algebras

2016 ◽  
Vol 2016 (715) ◽  
Author(s):  
Bart De Bruyn ◽  
Hendrik Van Maldeghem
Keyword(s):  
Division Algebra ◽  
Division Algebras ◽  
Polar Spaces ◽  
Dual Polar Spaces

AbstractSuppose 𝕆 is an alternative division algebra that is quadratic over some subfield 𝕂 of its center

scholarly journals MAL'CEV–NEUMANN RINGS AND NONCROSSED PRODUCT DIVISION ALGEBRAS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250052 ◽  
Author(s):  
CÉCILE COYETTE
Keyword(s):  
Division Algebra ◽  
Laurent Series ◽  
Elementary Proof ◽  
Formal Series ◽  
Direct Approach ◽  
Division Algebras ◽  
Sufficient Condition ◽  
Series Ring

The first section of this paper yields a sufficient condition for a Mal'cev–Neumann ring of formal series to be a noncrossed product division algebra. This result is used in Sec. 2 to give an elementary proof of the existence of noncrossed product division algebras (of degree 8 or degree p2 for p any odd prime). The arguments are based on those of Hanke in [A direct approach to noncrossed product division algebras, thesis dissertation, Postdam (2001), An explicit example of a noncrossed product division algebra, Math. Nachr.251 (2004) 51–68, A twisted Laurent series ring that is a noncrossed product, Israel. J. Math.150 (2005) 199–2003].

scholarly journals On projective characters of prime degree

1991 ◽  
Vol 33 (3) ◽  
pp. 311-321 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Prime Number ◽  
Structural Information ◽  
Alternating Group ◽  
Complex Numbers ◽  
Prime Degree ◽  
Counter Example ◽  
Projective Representations

All groups G considered in this paper are finite and all representations of G are defined over the field of complex numbers. The reader unfamiliar with projective representations is referred to [9] for basic definitions and elementary results.Let Proj(G, α) denote the set of irreducible projective characters of a group G with cocycle α. In a previous paper [3] the author showed that if G is a (p, α)-group, that is the degrees of the elements of Proj(G, α) are all powers of a prime number p, then G is solvable. However Isaacs and Passman in [8] were able to give structural information about a group G for which ξ(1) divides pe for all ξ ∈ Proj(G, 1), where 1 denotes the trivial cocycle of G, and indeed classified all such groups in the case e = l. Their results rely on the fact that G has a normal abelian p-complement, which is false in general if G is a (p, α)-group; the alternating group A4 providing an easy counter-example for p = 2.

Congruence of Ankeny–Artin–Chowla type for cyclic fields of prime degree l

1996 ◽  
Vol 119 (1) ◽  
pp. 17-22 ◽  
Author(s):  
Stanislav Jakubec
Keyword(s):  
Prime Number ◽  
Class Number ◽  
Quadratic Field ◽  
Fundamental Unit ◽  
Prime Degree ◽  
Image Position

Ankeny–Artin–Chowla obtained several congruences for the class number hk of a quadratic field K, some of which were also obtained by Kiselev. In particular, if the discriminant of K is a prime number p ≡ 1 (mod 4) and ε = t + u √p/2 is the fundamental unit of K, then

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