More on imaginaries in p-adic fields

1997 ◽  
Vol 62 (1) ◽  
pp. 1-13
Author(s):  
Philip Scowcroft
Keyword(s):  
Natural Number ◽  
Equivalence Relation ◽  
Real Closed Field ◽  
Complete Theory ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Image Position ◽  
Equivalent Conditions

According to [4, p. 1154], a complete L-theory T eliminates imaginaries just in case for every L-formula φ(x1,… , xm, y1, …, yn), every model M of T, and every ā Є Mn, there is a subset A of M's domain with the following property: if N ≽ M and f is an automorphism of N, thenif and only ifAmong the several equivalent conditions discussed in [4, p. 1155], one may single out the following: if T is a complete theory in which two distinct objects are definable, T eliminates imaginaries just in case every T-definable n-ary equivalence relation may be defined by a formulawhere g is a T-definable n-ary function taking k-tuples as values (for some natural number k).Say that an L-structure M eliminates imaginaries just in case Th(M) does. If L is the language of rings with unit, [4, p. 1158] shows that any algebraically closed field eliminates imaginaries, and [2, p. 629] points out that any real-closed field eliminates imaginaries.

Equations and Complexity for the Dubois–Efroymson Dimension Theorem

2009 ◽  
Vol 52 (2) ◽  
pp. 224-236
Author(s):  
Riccardo Ghiloni
Keyword(s):  
Real Closed Field ◽  
Closed Field ◽  
Algebraic Subset ◽  
Algebraically Closed Field ◽  
Algebraic Set ◽  
Real Algebraic Set ◽  
Nonsingular Point ◽  
Minimum Integer ◽  
The Ideal

AbstractLetRbe a real closed field, letX⊂Rnbe an irreducible real algebraic set and letZbe an algebraic subset ofXof codimension ≥ 2. Dubois and Efroymson proved the existence of an irreducible algebraic subset ofXof codimension 1 containingZ. We improve this dimension theorem as follows. Indicate by μ the minimum integer such that the ideal of polynomials inR[x1, … ,xn] vanishing onZcan be generated by polynomials of degree ≤ μ. We prove the following two results: (1) There exists a polynomialP∈R[x1, … ,xn] of degree≤ μ+1 such thatX∩P–1(0) is an irreducible algebraic subset ofXof codimension 1 containingZ. (2) LetFbe a polynomial inR[x1, … ,xn] of degreedvanishing onZ. Suppose there exists a nonsingular pointxofXsuch thatF(x) = 0 and the differential atxof the restriction ofFtoXis nonzero. Then there exists a polynomialG∈R[x1, … ,xn] of degree ≤ max﹛d, μ + 1﹜ such that, for eacht∈ (–1, 1) \ ﹛0﹜, the set ﹛x∈X|F(x) +tG(x) = 0﹜ is an irreducible algebraic subset ofXof codimension 1 containingZ. Result (1) and a slightly different version of result (2) are valid over any algebraically closed field also.

scholarly journals An algebraically closed field

1968 ◽  
Vol 9 (2) ◽  
pp. 146-151 ◽  
Author(s):  
F. J. Rayner
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Zero Characteristic ◽  
Image Position ◽  
Formal Power ◽  
Puiseux Expansions

Letkbe any algebraically closed field, and denote byk((t)) the field of formal power series in one indeterminatetoverk. Letso thatKis the field of Puiseux expansions with coefficients ink(each element ofKis a formal power series intl/rfor some positive integerr). It is well-known thatKis algebraically closed if and only ifkis of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous toKwhich is algebraically closed whenkhas non-zero characteristicp. In this paper, I prove that the setLof all formal power series of the form Σaitei(where (ei) is well-ordered,ei=mi|nprt,n∈ Ζ,mi∈ Ζ,ai∈k,ri∈ Ν) forms an algebraically closed field.

scholarly journals Involutions on finite-dimensional algebras over real closed fields

2004 ◽  
Vol 77 (1) ◽  
pp. 123-128 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Characteristic Zero ◽  
Real Closed Field ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
Real Closed Fields ◽  
Finite Dimensional ◽  
Closed Fields ◽  
Finite Dimensional Algebras

AbstractIt is shown that the following conditions on a finite-dimensional algebra A over a real closed field or an algebraically closed field of characteristic zero are equivalent: (i) A admits a special involution, in the sense of Easdown and Munn, (ii) A admits a proper involution, (iii) A is semisimple.

Model completeness of o-minimal structures expanded by Dedekind cuts

2005 ◽  
Vol 70 (1) ◽  
pp. 29-60 ◽  
Author(s):  
Marcus Tressl
Keyword(s):  
Valuation Ring ◽  
Ordered Set ◽  
Real Closed Field ◽  
Complete Theory ◽  
Closed Field ◽  
Model Completeness ◽  
Unary Predicate ◽  
Minimal Structure ◽  
Totally Ordered Set ◽  
Dedekind Cuts

§1. Introduction. Let M be a totally ordered set. A (Dedekind) cut p of M is a couple (pL, pR) of subsets pL, pR of M such that pL ⋃ pR = M and pL < pR, i.e., a < b for all a ∈ pL, b ∈ pR. In this article we are looking for model completeness results of o-minimal structures M expanded by a set pL for a cut p of M. This means the following. Let M be an o-minimal structure in the language L and suppose M is model complete. Let D be a new unary predicate and let p be a cut of (the underlying ordered set of) M. Then we are looking for a natural, definable expansion of the L(D)-structure (M, pL) which is model complete.The first result in this direction is a theorem of Cherlin and Dickmann (cf. [Ch-Dic]) which says that a real closed field expanded by a convex valuation ring has a model complete theory. This statement translates into the cuts language as follows. If Z is a subset of an ordered set M we write Z+ for the cut p with pR = {a ∈ M ∣ a > Z} and Z− for the cut q with qL = {a ∈ M ∣ a < Z}.

On Simple Connectedness of Minimal Representation-Infinite Algebras

2015 ◽  
Vol 22 (04) ◽  
pp. 639-654
Author(s):  
Hailou Yao ◽  
Guoqiang Han
Keyword(s):  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Closed Field ◽  
Minimal Representation ◽  
Algebraically Closed Field ◽  
Hochschild Cohomology Group ◽  
Simple Connectedness ◽  
Simply Connected ◽  
Equivalent Conditions ◽  
Infinite Algebras

Let A be a connected minimal representation-infinite algebra over an algebraically closed field k. In this paper, we investigate the simple connectedness and strong simple connectedness of A. We prove that A is simply connected if and only if its first Hochschild cohomology group H1(A) is trivial. We also give some equivalent conditions of strong simple connectedness of an algebra A.

On Matrix Commutators

1960 ◽  
Vol 12 ◽  
pp. 269-277 ◽  
Author(s):  
Marvin Marcus ◽  
Nisar A. Khan
Keyword(s):  
Linear Space ◽  
Elementary Divisor ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Elementary Divisors ◽  
Image Position

Let A, B, and X be n-square matrices over an algebraically closed field F of characteristic 0. Let [A, B] = AB — BA and set (A, B) = [A, [A, B]]. Recently several proofs (1; 3; 5) of the following result have appeared: if det (AB) ≠ 0 and (A,B) = 0 then A-1B-1AB - I is nilpotent. In (2) McCoy determined the general form of any X satisfying1.1in the case that A has a single elementary divisor corresponding to each eigenvalue, that is, A is non-derogatory. In Theorem 1 we determine the structure of any matrix X satisfying (1.1) and also give a formula for the dimension of the linear space of all such X in terms of the degrees of the elementary divisors of A.

Linear Relations Between Higher Matrix Commutators

1964 ◽  
Vol 16 ◽  
pp. 315-320 ◽  
Author(s):  
Nisar A. Khan
Keyword(s):  
Linear Relation ◽  
Closed Field ◽  
Research Papers ◽  
Algebraically Closed Field ◽  
Linear Relations ◽  
Image Position ◽  
Iterated Commutators

Let Mn denote the space of all n-square matrices over an algebraically closed field F. For A, B ∊ Mn, letdefine the iterated commutators of A and B. Recently several research papers (1, 2, 4, and 5) have appeared on these commutators. In (1), Kato and Taussky have proved that for n = 2 the iterated commutators of A and B satisfy the linear relation

COMPARING TWO VERSIONS OF THE REALS

2016 ◽  
Vol 81 (3) ◽  
pp. 1115-1123
Author(s):  
G. IGUSA ◽  
J. F. KNIGHT
Keyword(s):  
Residue Field ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Numbers ◽  
Computing Power ◽  
Ordered Field ◽  
Image Position ◽  
Recursively Saturated

AbstractSchweber [10] defined a reducibility that allows us to compare the computing power of structures of arbitrary cardinality. Here we focus on the ordered field ${\cal R}$ of real numbers and a structure ${\cal W}$ that just codes the subsets of ω. In [10], it was observed that ${\cal W}$ is reducible to ${\cal R}$. We prove that ${\cal R}$ is not reducible to ${\cal W}$. As part of the proof, we show that for a countable recursively saturated real closed field ${\cal P}$ with residue field k, some copy of ${\cal P}$ does not compute a copy of k.

The self-intersection formula and the ‘formule-clef’

1975 ◽  
Vol 78 (1) ◽  
pp. 117-123 ◽  
Author(s):  
A. T. Lascu ◽  
D. Mumford ◽  
D. B. Scott
Keyword(s):  
Ring Homomorphism ◽  
Equivalence Classes ◽  
The Self ◽  
Group Homomorphism ◽  
Closed Field ◽  
Chow Ring ◽  
Algebraically Closed Field ◽  
Rational Equivalence ◽  
Image Position ◽  
Intersection Formula

We shall consider exclusively algebraic non-singular quasi-projective irreducible varieties over an algebraically closed field. If V is such a variety will be the Chow ring of rational equivalence classes of cycles of Vand the group homomorphism defined by any proper morphism φ: V1 → V2. Alsodenotes the ring homomorphism defined by φ.

A Multivariable form of the Fundamental Theorem of Algebra

1983 ◽  
Vol 26 (3) ◽  
pp. 271-272
Author(s):  
Pablo M. Salzberg
Keyword(s):  
Fundamental Theorem ◽  
Homogeneous Polynomial ◽  
Inner Product ◽  
Closed Field ◽  
Sufficient Condition ◽  
Algebraically Closed Field ◽  
Fundamental Theorem Of Algebra ◽  
Image Position ◽  
Derivatives Of

AbstractLet H(x) be a homogeneous polynomial in n indeterminates over an algebraically closed field K. A necesssary and sufficient condition is given for H(x) to admit a factorization of the forma, b∈ Kn, and “∘” is the usual inner product. This condition involves the linear derivatives of H(x).

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