Bill Jacob

Division algebras over Henselian fields

In this chapter we focus on the tame division algebras D with center a field F with Henselian valuation v. As usual, we approach this by first obtaining results for graded division algebras, then lifting back from \(\operatorname {\mathsf {gr}}(D)\) to D. This is facilitated by results in §8.1 on existence and uniqueness of lifts of tame subalgebras from \(\operatorname {\mathsf {gr}}(D)\) to D. In §8.2, we describe four fundamental canonical (up to conjugacy) subalgebras of D that reflect its valuative structure. The rest of the chapter is devoted to Brauer group factorizations of D corresponding to the noncanonical direct product decomposition of \(\operatorname {\mathit{Br}}_{t}(F)\) given in Cor. 7.85. The factor \(\operatorname {\mathit{Hom}}^{c}(\operatorname {\mathcal {G}}(\overline{F}), {\mathbb{T}}(\Gamma_{F}))\) is represented by a type of division algebra N called decomposably semiramified, defined in §8.3, and characterized by the property that N contains a maximal subfield inertial over F and another totally ramified over F. We show in §8.4 that every tame division algebra D is Brauer equivalent to some S⊗ F T where S is inertially split and T is tame and totally ramified over F. We show further that every inertially split division algebra S is Brauer equivalent to some I⊗ F N, where I is inertial over F and N is decomposably semiramified. The classes \([\,\overline{I}\,]\) for the I appearing in the I⊗ F N decompositions of S are shown to range over a single coset of \(\mathit{Dec}(Z(\overline{S})/\overline{F})\) in \(\operatorname {\mathit{Br}}(\overline{F})\), called the specialization coset of S. In the final subsection, §8.4.6, we summarize what happens in the special case that v is discrete of rank 1, where substantial simplifications occur.

Rigid Elements, Valuations, and Realization of Witt Rings

to the theory over the residue class field of the valuation. This theorem generalizes to any 2-henselian valuation (cf. [Kl, Sect. 12.21). Let v be the valuation. Let a E k := F\{O 1 be such that v(a) is not divisible by two. The key idea behind the proof is that the quadratic form (1, a} only represents elements in % u a%. Such an element a is called

On quadratic forms and Galois cohomology

Witt rings. Specifically, suppose there is a morphism of abstract Witt rings W F —> R. Does there exist a field extension K of F such that WK ~ R and the given morphism corresponds to %KjF W F —> WK? A classical example is the case R ~ Z . Then K can be chosen to be a real closure of F relative to the ordering induced by the morphism. Unfortunately, the answer is in general negative. But in the important case when W F = R x S in the category of abstract Witt rings and the morphism is the projection map, we have a positive result. In [7] we extend the work of [18] and [30] on valuations and use this extension to show THEOREM 8. Let (j> : W F^RxS be an isomorphism of abstract Witt rings and let 7r : R x S —> R be the projection. Assume that R is not basic (i.e., R is a group Witt ring over a subring). Then there exists a 2-extension K of F and an isomorphism ip : WK^R of abstract Witt rings such that the diagram WF — ^ RxS [ j * WK ——> R

Prospective Secondary Mathematics Teachers’ Understanding and Cognitive Difficulties in Making Connections among Representations

This study investigates prospective secondary teachers’ cognitive difficulties and mathematical ideas involved in making connections among representations. We implemented a three-week teaching unit to help prospective secondary mathematics teachers develop understanding of big ideas that are critical to formulating connections among representations, in the context of conic curves. Qualitative analysis of data showed that most undergraduate mathematics majors and minors in this study struggled with variation, the Cartesian Connection, and other affiliated ideas such as graph as a locus of points. Furthermore, they were unable to identify basic metric relations encoded in algebraic expressions such as the distance between points, which further compounded their difficulties in making connections among representations. We argue that mathematics teacher education needs more focus on these ideas so that their graduates can successfully teach these big ideas in their future instruction.

Relative Brauer groups in characteristic

This paper gives a description of the relative Brauer group Br(E/F) when F has characteristic p, [E : F] = p, and the Galois group Gal(E 1 /F) is solvable, where E 1 is the Galois closure of E over F.

Division algebras with no common subfields

An example is given of division algebrasD1 andD2 of odd prime degreep over a fieldK such thatD1 andD2 have no common subfield properly containingF, butD1⊗i ⊗KD2 is not a division algebra for 1≤i≤p−1.

On profinite spaces of orderings

In this paper we present the following two results: we give an explicit description of the space of orderings (XQ(x),GQ(x)) as an inverse limit of finite spaces of orderings and we provide a new, simple proof of the fact that the class of spaces of orderings for which the pp conjecture holds true is closed under inverse limits. We discuss how these theorems interact with each other, and explain our motivation to look into these problems.

Partial Fractions and Milnor K-Theory

This paper uses partial fraction decompositions to give a direct computation of the logarithmic derivative of the norm in Milnor K-theory for a finite separable extension. This result is useful for computations involving the relative Brauer group in finite characteristic and Witt kernels for function fields in characteristic two. Katos result that the norm is compatible with the trace under logarithmic differentiation also follows from these tools. When F(x) is rational over F in finite characteristic ℓ, the unramified part of is computed to be .

The Witt ring of an elliptic curve over a local field

J6n Kr. Arason l, Richard Elman z,*, Bill Jacob 3,* 1 Raunvisindastofnun Hfisk61ans, University of Iceland, Reykjavik, Iceland (e-mail: jka@rhi.hi.is) 2 Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095, USA (e-mail: rse@math.ucla.edu) -~ Department of Mathematics, University of Califomiat, Santa Barbara, Santa Barbara, CA 93106, USA (e-mail: jacob@rnath.ncsb.edu)

Matrices as Linear Transformations

We all learned at an early age that in spite of the similarities, there is a significant difference between a left shoe and a right shoe. How does a mathematician recognize and describe this difference? To a mathematician, the right shoe is the reflection of the left shoe, and vice versa. Look in the mirror at a left shoe’s reflection next to its companion right shoe to see why. It turns out that the difference between some objects and their reflections can be substantial, even more so than for shoes!