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Автор: Grafiati

Опубліковано: 4 червня 2021

Оновлено: 5 березня 2023

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1

JARDEN, MOSHE, and ALEXANDRA SHLAPENTOKH. "DECIDABLE ALGEBRAIC FIELDS." Journal of Symbolic Logic 82, no. 2 (June 2017): 474–88. http://dx.doi.org/10.1017/jsl.2017.10.

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AbstractWe discuss the connection between decidability of a theory of a large algebraic extensions of ${\Bbb Q}$ and the recursiveness of the field as a subset of a fixed algebraic closure. In particular, we prove that if an algebraic extension K of ${\Bbb Q}$ has a decidable existential theory, then within any fixed algebraic closure $\widetilde{\Bbb Q}$ of ${\Bbb Q}$, the field K must be conjugate over ${\Bbb Q}$ to a field which is recursive as a subset of the algebraic closure. We also show that for each positive integer e there are infinitely many e-tuples $\sigma \in {\text{Gal}}\left( {\Bbb Q} \right)^e $ such that the field $\widetilde{\Bbb Q}\left( \sigma \right)$ is primitive recursive in $\widetilde{\Bbb Q}$ and its elementary theory is primitive recursively decidable. Moreover, $\widetilde{\Bbb Q}\left( \sigma \right)$ is PAC and ${\text{Gal}}\left( {\widetilde{\Bbb Q}\left( \sigma \right)} \right)$ is isomorphic to the free profinite group on e generators.

2

Kuz'min, L. V. "Algebraic number fields." Journal of Soviet Mathematics 38, no. 3 (August 1987): 1930–88. http://dx.doi.org/10.1007/bf01093434.

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3

Chudnovsky, D. V., and G. V. Chudnovsky. "Algebraic complexities and algebraic curves over finite fields." Journal of Complexity 4, no. 4 (December 1988): 285–316. http://dx.doi.org/10.1016/0885-064x(88)90012-x.

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4

Bost, Jean-Benoît. "Algebraic leaves of algebraic foliations over number fields." Publications mathématiques de l'IHÉS 93, no. 1 (September 2001): 161–221. http://dx.doi.org/10.1007/s10240-001-8191-3.

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5

Chudnovsky, D. V., and G. V. Chudnovsky. "Algebraic complexities and algebraic curves over finite fields." Proceedings of the National Academy of Sciences 84, no. 7 (April 1, 1987): 1739–43. http://dx.doi.org/10.1073/pnas.84.7.1739.

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6

Beyarslan, Özlem, and Ehud Hrushovski. "On algebraic closure in pseudofinite fields." Journal of Symbolic Logic 77, no. 4 (December 2012): 1057–66. http://dx.doi.org/10.2178/jsl.7704010.

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AbstractWe study the automorphism group of the algebraic closure of a substructureAof a pseudo-finite fieldF. We show that the behavior of this group, even whenAis large, depends essentially on the roots of unity inF. For almost all completions of the theory of pseudofinite fields, we show that overA, algebraic closure agrees with definable closure, as soon asAcontains the relative algebraic closure of the prime field.

7

Praeger, Cheryl E. "Kronecker classes of fields and covering subgroups of finite groups." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 57, no. 1 (August 1994): 17–34. http://dx.doi.org/10.1017/s1446788700036028.

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AbstractKronecker classes of algebraci number fields were introduced by W. Jehne in an attempt to understand the extent to which the structure of an extension K: k of algebraic number fields was influenced by the decomposition of primes of k over K. He found an important link between Kronecker equivalent field extensions and a certain covering property of their Galois groups. This surveys recent contributions of Group Theory to the understanding of Kronecker equivalence of algebraic number fields. In particular some group theoretic conjectures related to the Kronecker class of an extension of bounded degree are explored.

8

Junker, Markus, and Jochen Koenigsmann. "Schlanke Körper (Slim fields)." Journal of Symbolic Logic 75, no. 2 (June 2010): 481–500. http://dx.doi.org/10.2178/jsl/1268917491.

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AbstractWe examine fields in which model theoretic algebraic closure coincides with relative field theoretic algebraic closure. These are perfect fields with nice model theoretic behaviour. For example they are exactly the fields in which algebraic independence is an abstract independence relation in the sense of Kim and Pillay. Classes of examples are perfect PAC fields, model complete large fields and henselian valued fields of characteristic 0.

9

KEKEÇ, GÜLCAN. "-NUMBERS IN FIELDS OF FORMAL POWER SERIES OVER FINITE FIELDS." Bulletin of the Australian Mathematical Society 101, no. 2 (July 29, 2019): 218–25. http://dx.doi.org/10.1017/s0004972719000832.

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In the field $\mathbb{K}$ of formal power series over a finite field $K$, we consider some lacunary power series with algebraic coefficients in a finite extension of $K(x)$. We show that the values of these series at nonzero algebraic arguments in $\mathbb{K}$ are $U$-numbers.

10

Restuccia, Gaetana. "Algebraic models in different fields." Applied Mathematical Sciences 8 (2014): 8345–51. http://dx.doi.org/10.12988/ams.2014.411922.

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11

van Hoeij, Mark, and Vivek Pal. "Isomorphisms of algebraic number fields." Journal de Théorie des Nombres de Bordeaux 24, no. 2 (2012): 293–305. http://dx.doi.org/10.5802/jtnb.797.

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12

Kollár, János. "Algebraic varieties over PAC fields." Israel Journal of Mathematics 161, no. 1 (October 2007): 89–101. http://dx.doi.org/10.1007/s11856-007-0073-z.

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13

Popescu, Dorin. "Algebraic extensions of valued fields." Journal of Algebra 108, no. 2 (July 1987): 513–33. http://dx.doi.org/10.1016/0021-8693(87)90114-1.

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14

Grekhov, M. V. "Integral Models of Algebraic Tori Over Fields of Algebraic Numbers." Journal of Mathematical Sciences 219, no. 3 (October 24, 2016): 413–26. http://dx.doi.org/10.1007/s10958-016-3117-2.

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15

Scanlon, Thomas, and José Felipe Voloch. "Difference algebraic subgroups of commutative algebraic groups over finite fields." manuscripta mathematica 99, no. 3 (July 1, 1999): 329–39. http://dx.doi.org/10.1007/s002290050176.

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16

Scheicher, Klaus. "β-Expansions in algebraic function fields over finite fields". Finite Fields and Their Applications 13, № 2 (квітень 2007): 394–410. http://dx.doi.org/10.1016/j.ffa.2005.08.008.

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17

Andrade, Antonio A., Agnaldo J. Ferrari, José C. Interlando, and Robson R. Araujo. "Constructions of Dense Lattices over Number Fields." TEMA (São Carlos) 21, no. 1 (March 27, 2020): 57. http://dx.doi.org/10.5540/tema.2020.021.01.57.

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In this work, we present constructions of algebraic lattices in Euclidean space with optimal center density in dimensions 2,3,4,5,6,8 and 12, which are rotated versions of the lattices Lambda_n, for n =2,3,4,5,6,8 and K_12. These algebraic lattices are constructed through canonical homomorphism via Z-modules of the ring of algebraic integers of a number field.

18

He, Yang-Hui. "Fields over Fields." inSTEMM Journal 1, S1 (July 15, 2022): 15–46. http://dx.doi.org/10.56725/instemm.v1is1.9.

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We investigate certain arithmetic properties of field theories. In particular, we study the vacuum structure of supersymmetric gauge theories as algebraic varieties over number fields of finite characteristic. Parallel to the Plethystic Programme of counting the spectrum of operators from the syzygies of the complex geometry, we construct, based on the zeros of the vacuum moduli space over finite fields, the local and global Hasse-Weil zeta functions, as well as develop the associated Dirichlet expansions. We find curious dualities wherein the geometrical properties and asymptotic behaviour of one gauge theory is governed by the number theoretic nature of another.

19

Olson, Loren D. "Parametrized units in algebraic number fields." Mathematical Proceedings of the Cambridge Philosophical Society 103, no. 1 (January 1988): 15–25. http://dx.doi.org/10.1017/s0305004100064574.

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One of the fundamental problems in algebraic number theory is the construction of units in algebraic number fields. Various authors have considered number fields which are parametrized by an integer variable. They have described units in these fields by polynomial expressions in the variable e.g. the fields ℚ(√[N2 + 1]), Nεℤ, with the units εN = N + √[N2 + l]. We begin this article by formulating a general principle for obtaining units in algebraic function fields and candidates for units in parametrized families of algebraic number fields. We show that many of the cases considered previously in the literature by such authors as Bernstein [2], Neubrand [8], and Stender [ll] fall in under this principle. Often the results may be obtained much more easily than before. We then examine the connection between parametrized cubic fields and elliptic curves. In §4 we consider parametrized quadratic fields, a situation previously studied by Neubrand [8]. We conclude in §5 by examining the effect of parametrizing the torsion structure on an elliptic curve at the same time.

20

Miller, Russell. "d-computable categoricity for algebraic fields." Journal of Symbolic Logic 74, no. 4 (December 2009): 1325–51. http://dx.doi.org/10.2178/jsl/1254748694.

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AbstractWe use the Low Basis Theorem of Jockusch and Soare to show that all computable algebraic fields are d-computably categorical for a particular Turing degree d with d′ = 0″, but that not all such fields are 0′-computably categorical. We also prove related results about algebraic fields with splitting algorithms, and fields of finite transcendence degree over ℚ.

21

Narkiewicz, W. "Polynomial cycles in algebraic number fields." Colloquium Mathematicum 58, no. 1 (1989): 151–55. http://dx.doi.org/10.4064/cm-58-1-151-155.

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22

Luo, Zhaohua. "Kodaira Dimension of Algebraic Function Fields." American Journal of Mathematics 109, no. 4 (August 1987): 669. http://dx.doi.org/10.2307/2374609.

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23

Lettl, Günter. "Thue equations over algebraic function fields." Acta Arithmetica 117, no. 2 (2005): 107–23. http://dx.doi.org/10.4064/aa117-2-1.

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24

Kojima, Hisashi. "Galois extensions of algebraic function fields." Tohoku Mathematical Journal 42, no. 2 (1990): 149–61. http://dx.doi.org/10.2748/tmj/1178227651.

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25

Chen, Lu, and Tobias Fritz. "An algebraic approach to physical fields." Studies in History and Philosophy of Science Part A 89 (October 2021): 188–201. http://dx.doi.org/10.1016/j.shpsa.2021.08.011.

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26

Gładki, Paweł, and Mateusz Pulikowski. "Gauss Congruences in Algebraic Number Fields." Annales Mathematicae Silesianae 36, no. 1 (January 17, 2022): 53–56. http://dx.doi.org/10.2478/amsil-2022-0002.

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27

Waterman, P. L., and C. Maclachlan. "Fuchsian groups and algebraic number fields." Transactions of the American Mathematical Society 287, no. 1 (January 1, 1985): 353. http://dx.doi.org/10.1090/s0002-9947-1985-0766224-5.

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28

Hirschfeldt, Denis R., Ken Kramer, Russell Miller, and Alexandra Shlapentokh. "Categoricity properties for computable algebraic fields." Transactions of the American Mathematical Society 367, no. 6 (October 20, 2014): 3981–4017. http://dx.doi.org/10.1090/s0002-9947-2014-06094-7.

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29

Sancho de Salas, Juan B. "Tangent algebraic subvarieties of vector fields." Transactions of the American Mathematical Society 357, no. 9 (October 7, 2004): 3509–23. http://dx.doi.org/10.1090/s0002-9947-04-03584-6.

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30

Semaev, Igor. "Sparse Algebraic Equations over Finite Fields." SIAM Journal on Computing 39, no. 2 (January 2009): 388–409. http://dx.doi.org/10.1137/070700371.

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31

Forbes, G. W., D. J. Butler, R. L. Gordon, and A. A. Asatryan. "Algebraic corrections for paraxial wave fields." Journal of the Optical Society of America A 14, no. 12 (December 1, 1997): 3300. http://dx.doi.org/10.1364/josaa.14.003300.

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32

Dixon, John D. "Computing subfields in algebraic number fields." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 49, no. 3 (December 1990): 434–48. http://dx.doi.org/10.1017/s1446788700032432.

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AbstractLet K:= Q(α) be an algebraic number field which is given by specifying the minimal polynomial f(X) for α over Q. We describe a procedure for finding the subfields L of K by constructing pairs (w(X), g(X)) of polynomials over Q such that L= Q(w(α)) and g(X) is the minimal polynomial for w(α). The construction uses local information obtained from the Frobenius-Chebotarev theorem about the Galois group Gal(f), and computations over p-adic extensions of Q.

33

Emiris, Ioannis Z., Angelos Mantzaflaris, and Bernard Mourrain. "Voronoi diagrams of algebraic distance fields." Computer-Aided Design 45, no. 2 (February 2013): 511–16. http://dx.doi.org/10.1016/j.cad.2012.10.043.

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34

Lu, M., D. Wan, L. P. Wang, and X. D. Zhang. "Algebraic Cayley graphs over finite fields." Finite Fields and Their Applications 28 (July 2014): 43–56. http://dx.doi.org/10.1016/j.ffa.2014.01.014.

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35

Abouzahra, M., and L. Lewin. "The polylogarithm in algebraic number fields." Journal of Number Theory 21, no. 2 (October 1985): 214–44. http://dx.doi.org/10.1016/0022-314x(85)90052-6.

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36

Haran, Dan. "Hilbertian fields under separable algebraic extensions." Inventiones Mathematicae 137, no. 1 (June 1, 1999): 113–26. http://dx.doi.org/10.1007/s002220050325.

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37

INGRAM, PATRICK, VALÉRY MAHÉ, JOSEPH H. SILVERMAN, KATHERINE E. STANGE, and MARCO STRENG. "ALGEBRAIC DIVISIBILITY SEQUENCES OVER FUNCTION FIELDS." Journal of the Australian Mathematical Society 92, no. 1 (February 2012): 99–126. http://dx.doi.org/10.1017/s1446788712000092.

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AbstractIn this note we study the existence of primes and of primitive divisors in function field analogues of classical divisibility sequences. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.

38

Fein, Burton, and Murray Schacher. "Brauer groups of algebraic function fields." Journal of Algebra 103, no. 2 (October 1986): 454–65. http://dx.doi.org/10.1016/0021-8693(86)90146-8.

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39

Coutinho, S. C., and L. Menasché Schechter. "Algebraic solutions of plane vector fields." Journal of Pure and Applied Algebra 213, no. 1 (January 2009): 144–53. http://dx.doi.org/10.1016/j.jpaa.2008.06.003.

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40

Endo, Shizuo, and Ming-Chang Kang. "Function fields of algebraic tori revisited." Asian Journal of Mathematics 21, no. 2 (2017): 197–224. http://dx.doi.org/10.4310/ajm.2017.v21.n2.a1.

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41

Palmer, David, David Bommes, and Justin Solomon. "Algebraic Representations for Volumetric Frame Fields." ACM Transactions on Graphics 39, no. 2 (April 14, 2020): 1–17. http://dx.doi.org/10.1145/3366786.

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42

Landau, Susan. "Factoring Polynomials over Algebraic Number Fields." SIAM Journal on Computing 14, no. 1 (February 1985): 184–95. http://dx.doi.org/10.1137/0214015.

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43

Fein, B., and M. Schacher. "Crossed Products over Algebraic Function Fields." Journal of Algebra 171, no. 2 (January 1995): 531–40. http://dx.doi.org/10.1006/jabr.1995.1026.

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44

Rausch, U. "Character Sums in Algebraic Number Fields." Journal of Number Theory 46, no. 2 (February 1994): 179–95. http://dx.doi.org/10.1006/jnth.1994.1011.

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45

Philip, G. M., C. Gregory Skilbeck, and D. F. Watson. "Algebraic dispersion fields on ternary diagrams." Mathematical Geology 19, no. 3 (April 1987): 171–81. http://dx.doi.org/10.1007/bf00897745.

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46

Shapiro, Harold N., and Alexandra Shlapentokh. "Diophantine relationships between algebraic number fields." Communications on Pure and Applied Mathematics 42, no. 8 (December 1989): 1113–22. http://dx.doi.org/10.1002/cpa.3160420805.

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47

Khanduja, Sudesh K. "The discriminant of compositum of algebraic number fields." International Journal of Number Theory 15, no. 02 (March 2019): 353–60. http://dx.doi.org/10.1142/s1793042119500167.

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Анотація:

For an algebraic number field [Formula: see text], let [Formula: see text] denote the discriminant of an algebraic number field [Formula: see text]. It is well known that if [Formula: see text] are algebraic number fields with coprime discriminants, then [Formula: see text] are linearly disjoint over the field [Formula: see text] of rational numbers and [Formula: see text], [Formula: see text] being the degree of [Formula: see text] over [Formula: see text]. In this paper, we prove that the converse of this result holds in relative extensions of algebraic number fields. We also give some more necessary and sufficient conditions for the analogue of the above equality to hold for algebraic number fields [Formula: see text] linearly disjoint over [Formula: see text].

48

Binyamini, Gal. "Bezout-type theorems for differential fields." Compositio Mathematica 153, no. 4 (March 13, 2017): 867–88. http://dx.doi.org/10.1112/s0010437x17007035.

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We prove analogs of the Bezout and the Bernstein–Kushnirenko–Khovanskii theorems for systems of algebraic differential conditions over differentially closed fields. Namely, given a system of algebraic conditions on the first $l$ derivatives of an $n$-tuple of functions, which admits finitely many solutions, we show that the number of solutions is bounded by an appropriate constant (depending singly-exponentially on $n$ and $l$) times the volume of the Newton polytope of the set of conditions. This improves a doubly-exponential estimate due to Hrushovski and Pillay. We illustrate the application of our estimates in two diophantine contexts: to counting transcendental lattice points on algebraic subvarieties of semi-abelian varieties, following Hrushovski and Pillay; and to counting the number of intersections between isogeny classes of elliptic curves and algebraic varieties, following Freitag and Scanlon. In both cases we obtain bounds which are singly-exponential (improving the known doubly-exponential bounds) and which exhibit the natural asymptotic growth with respect to the degrees of the equations involved.

49

Kaliman, Shulim, Frank Kutzschebauch, and Matthias Leuenberger. "Complete algebraic vector fields on affine surfaces." International Journal of Mathematics 31, no. 03 (January 14, 2020): 2050018. http://dx.doi.org/10.1142/s0129167x20500184.

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Анотація:

Let [Formula: see text] be the subgroup of the group [Formula: see text] of holomorphic automorphisms of a normal affine algebraic surface [Formula: see text] generated by elements of flows associated with complete algebraic vector fields. Our main result is a classification of all normal affine algebraic surfaces [Formula: see text] quasi-homogeneous under [Formula: see text] in terms of the dual graphs of the boundaries [Formula: see text] of their SNC-completions [Formula: see text].

50

Pillay, Anand. "Differentially algebraic group chunks." Journal of Symbolic Logic 55, no. 3 (September 1990): 1138–42. http://dx.doi.org/10.2307/2274479.

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Анотація:

We point out that a group first order definable in a differentially closed field K of characteristic 0 can be definably equipped with the structure of a differentially algebraic group over K. This is a translation into the framework of differentially closed fields of what is known for groups definable in algebraically closed fields (Weil's theorem).I restrict myself here to showing (Theorem 20) how one can find a large “differentially algebraic group chunk” inside a group defined in a differentially closed field. The rest of the translation (Theorem 21) follows routinely, as in [B].What is, perhaps, of interest is that the proof proceeds at a completely general (soft) model theoretic level, once Facts 1–4 below are known.Fact 1. The theory of differentially closed fields of characteristic 0 is complete and has quantifier elimination in the language of differential fields (+, ·,0,1, −1,d).Fact 2. Affine n-space over a differentially closed field is a Noetherian space when equipped with the differential Zariski topology.Fact 3. If K is a differentially closed field, k ⊆ K a differential field, and a and are in k, then a is in the definable closure of k ◡ iff a ∈ ‹› (where k ‹› denotes the differential field generated by k and).Fact 4. The theory of differentially closed fields of characteristic zero is totally transcendental (in particular, stable).