Quintic Forms overp-adic Fields

AbstractWe prove that a quintic form in 26 variables defined over ap-adic fieldKalways has a nontrivial zero overKif the residue class field ofKhas at least 47 elements. This is in agreement with the theorem of Ax–Kochen which states that a homogeneous form of degreedind2+1 variables defined overQphas a nontrivialQp-rational zero ifpis sufficiently large. The Ax–Kochen theorem gives no results on the bound forp. Ford=1, 2, 3 it has been known for a long time that there is a nontrivialQp-rational zero for all values ofp. Ford=4, Terjanian gave an example of a form in 18 variables overQ2having no nontrivialQ2-rational zero. This is the first result which gives an effective bound for the cased=5

Triple linkage of quadratic Pfister forms

Given a field F of characteristic 2, we prove that if every three quadratic nfold Pfister forms have a common quadratic (n - 1)- fold Pfister factor then I n+ 1 q 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 char(F) = 2 and u(F) = 4, then every three quaternion algebras share a common maximal subfield

The Chemical Compositions of the SRd Variable Stars-- II. WY Andromedae, VW Eridani, and UW Librae

Chemical compositions are derived from high-resolution spectra for three stars classed as SRd variables in the General Catalogue of Variable Stars. These stars are shown to be metal-poor supergiants: WY And with [Fe/H] = -1.0, VW Eri with [Fe/H] = -1.8, and UW Lib with [Fe/H] = -1.2. Their compositions are identical to within the measurement errors with the compositions of subdwarfs, subgiants, and less evolved giants of the same FeH. The stars are at the tip of the first giant branch or in the early stages of evolution along the asymptotic giant branch (AGB). There is no convincing evidence that these SRd variables are experiencing thermal pulsing and the third dredge-up on the AGB. The SRds appear to be the cool limit of the sequence of RV Tauri variables.Comment: 14 pages, 1 figure, 4 table

The Elman-Lam-Krüskemper theorem

For a (formally) real field K, the vanishing of a certain power of the fundamental ideal in the Witt ring of K(√-1) implies that the same power of the fundamental ideal in the Witt ring of K is torsion free. The proof of this statement involves a fact on the structure of the torsion part of powers of the fundamental ideal in the Witt ring of K. This fact is very difficult to prove in general, but has an elementary proof under an assumption on the stability index of K. We present an exposition of these results