scholarly journals Zeros of systems of 𝔭-adic quadratic forms

2010 ◽  
Vol 146 (2) ◽  
pp. 271-287 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Algebraic Geometry ◽  
Quadratic Forms ◽  
Residue Class ◽  
Residue Field ◽  
Local Fields ◽  
Class Field ◽  
Residue Class Field ◽  
Minimization Technique ◽  
Diophantine Problems

AbstractWe show that a system of r quadratic forms over a 𝔭-adic field in at least 4r+1 variables will have a non-trivial zero as soon as the cardinality of the residue field is large enough. In contrast, the Ax–Kochen theorem [J. Ax and S. Kochen, Diophantine problems over local fields. I, Amer. J. Math. 87 (1965), 605–630] requires the characteristic to be large in terms of the degree of the field over ℚp. The proofs use a 𝔭-adic minimization technique, together with counting arguments over the residue class field, based on considerations from algebraic geometry.

scholarly journals The equation y′ = fy in zero residue characteristic

1991 ◽  
Vol 33 (2) ◽  
pp. 149-153
Author(s):  
Alain Escassut ◽  
Marie-Claude Sarmant
Keyword(s):  
Uniform Convergence ◽  
Characteristic Zero ◽  
Residue Class ◽  
Rational Functions ◽  
Closed Field ◽  
Class Field ◽  
Absolute Value ◽  
Residue Class Field ◽  
Analytic Elements ◽  
Residue Characteristic

Let K be an algebraically closed field complete with respect to an ultrametric absolute value |.| and let k be its residue class field. We assume k to have characteristic zero (hence K has characteristic zero too).Let D be a clopen bounded infraconnected set [3] in K, let R(D) be the algebra of the rational functions with no pole in D, let ‖.‖D be the norm of uniform convergence on D defined on R(D), and let H(D) be the algebra of the analytic elements on D i.e. the completion of R(D) for the norm ‖.‖D.

A lifting theorem for modular representations

1959 ◽  
Vol 252 (1268) ◽  
pp. 135-142 ◽  
Keyword(s):  
Residue Class ◽  
Sufficient Conditions ◽  
Modular Representations ◽  
Class Field ◽  
Residue Class Field ◽  
Homological Invariants

k is the residue-class field o/p of a ring 0 of p-adic integers. Sufficient conditions are found, I that a given representation p of a group G over k may be lifted to a representation r of G over o, particularly in the case where it is assumed that such a lifting exists for the restriction of p to a given subgroup H of G . The conditions involve certain homological invariants of p .

scholarly journals Direct Summands of Syzygy Modules of the Residue Class Field

2008 ◽  
Vol 189 ◽  
pp. 1-25 ◽  
Author(s):  
Ryo Takahashi
Keyword(s):  
Direct Summand ◽  
Local Ring ◽  
Residue Class ◽  
Class Field ◽  
Noetherian Local Ring ◽  
Residue Class Field ◽  
Syzygy Module ◽  
Syzygy Modules

AbstractLet R be a commutative Noetherian local ring. This paper deals with the problem asking whether R is Gorenstein if the nth syzygy module of the residue class field of R has a non-trivial direct summand of finite G-dimension for some n. It is proved that if n is at most two then it is true, and moreover, the structure of the ring R is determined essentially uniquely.

scholarly journals Ramification Theory for Extensions of Degree p

1971 ◽  
Vol 41 ◽  
pp. 149-168 ◽  
Author(s):  
Susan Williamson
Keyword(s):  
Residue Class ◽  
Field Extension ◽  
Quotient Field ◽  
Class Field ◽  
Wild Ramification ◽  
Valuation Rings ◽  
Ramification Theory ◽  
Residue Class Field ◽  
Rank One

The notions of tame and wild ramification lead us to make the following definition.Definition. The quotient field extension of an extension of discrete rank one valuation rings is said to be fiercely ramified if the residue class field extension has a nontrivial inseparable part.

Knowability

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Residue Class ◽  
Class Ii ◽  
Gauss Sums ◽  
Class I ◽  
Class Field ◽  
Residue Class Field

This chapter introduces the Knowability Lemma, which explains when products of Gauss sums associated to elements of a preaccordion are explicitly evaluable as polynomials in q, the order of the residue class field. It considers an episode in the cartoon associated to the short Gelfand-Tsetlin pattern and the three cases that apply according to the Knowability Lemma, two of which are maximality and knowability. Knowability is not important for the proof that Statement C implies Statement B. The chapter discusses the cases where ε‎ is Class II or Class I, leaving the remaining two cases to the reader. It also describes the variant of the argument for the case that ε‎ is of Class I, again leaving the two other cases to the reader.

Relative Brauer group and pro-p Galois group of pre-p-henselian fields

2015 ◽  
Vol 14 (06) ◽  
pp. 1550087
Author(s):  
R. P. Dario ◽  
A. J. Engler
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Residue Class ◽  
Brauer Group ◽  
Valued Fields ◽  
Class Field ◽  
Valued Field ◽  
Residue Class Field ◽  
Symbol Algebra ◽  
Torsion Part

Let p be a prime number and (F, v) a valued field. In this paper, we find a presentation for the p-torsion part of the Brauer group Br (F), by means of the valuation v. We only assume that F has a primitive pth root of the unity and the residue class field has characteristic not equal to p. This result naturally leads to consider valued fields that we call pre-p-henselian fields. It concerns valuations compatible with Rp, the p-radical of the field. To be precise, Rp is the radical of the skew-symmetric pairing which associates to a pair (a, b) the class of the symbol algebra (F; a, b) in Br F. In our main result, we state that pre-p-henselian fields are precisely the fields for which the Galois group of the maximal Galois p-extension admits a particular decomposition as a free pro-p product.

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