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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.
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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.
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.
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.
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.
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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.
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.
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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.
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.
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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.
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.
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Restuccia, Gaetana. "Algebraic models in different fields." Applied Mathematical Sciences 8 (2014): 8345–51. http://dx.doi.org/10.12988/ams.2014.411922.
This thesis investigates various theorems on polynomials over the rationals, algebraic numbers, algebraic integers, and quadratic fields. The material selected in this study is more of a number theoretical aspect than that of an algebraic structural aspect. Therefore, the topics of divisibility, unique factorization, prime numbers, and the roots of certain polynomials have been chosen for primary consideration.
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Lötter, Ernest C. "On towers of function fields over finite fields /." Link to the online version, 2007. http://hdl.handle.net/10019.1/1283.
Swanson, Colleen M. "Algebraic number fields and codes /." Connect to online version, 2006. http://ada.mtholyoke.edu/setr/websrc/pdfs/www/2006/172.pdf.
Rovi, Carmen. "Algebraic Curves over Finite Fields." Thesis, Linköping University, Department of Mathematics, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-56761.
This thesis surveys the issue of finding rational points on algebraic curves over finite fields. Since Goppa's construction of algebraic geometric codes, there has been great interest in finding curves with many rational points. Here we explain the main tools for finding rational points on a curve over a nite eld and provide the necessary background on ring and field theory. Four different articles are analyzed, the first of these articles gives a complete set of table showing the numbers of rational points for curves with genus up to 50. The other articles provide interesting constructions of covering curves: covers by the Hemitian curve, Kummer extensions and Artin-Schreier extensions. With these articles the great difficulty of finding explicit equations for curves with many rational points is overcome. With the method given by Arnaldo García in [6] we have been able to nd examples that can be used to define the lower bounds for the corresponding entries in the tables given in http: //wins.uva.nl/~geer, which to the time of writing this Thesis appear as "no information available". In fact, as the curves found are maximal, these entries no longer need a bound, they can be given by a unique entry, since the exact value of Nq(g) is now known.
At the end of the thesis an outline of the construction of Goppa codes is given and the NXL and XNL codes are presented.
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Rozario, Rebecca. "The Distribution of the Irreducibles in an Algebraic Number Field." Fogler Library, University of Maine, 2003. http://www.library.umaine.edu/theses/pdf/RozarioR2003.pdf.
The theory of operads is a conceptual framework that has become a kind of universal language, relating branches of topology and algebra. This thesis uses the operadic framework to study the derived algebraic properties of polyvector fields on manifolds.The thesis is divided into eight chapters. The first is an introduction to the thesis and the research field to which it belongs, while the second chapter surveys the basic mathematical results of the field.The third chapter is devoted to a novel construction of differential graded operads, generalizing an earlier construction due to Thomas Willwacher. The construction highlights and explains several categorical properties of differential graded algebras (of some kind) that come equipped with an action by a differential graded Lie algebra. In particular, the construction clarifies the deformation theory of such algebras and explains how such algebras can be twisted by Maurer-Cartan elements.The fourth chapter constructs an explicit strong homotopy deformation of polynomial polyvector fields on affine space, regarded as a two-colored noncommutative Gerstenhaber algebra. It also constructs an explicit strong homotopy quasi-isomorphism from this deformation to the canonical two-colored noncommmutative Gerstenhaber algebra of polydifferential operators on the affine space. This explicit construction generalizes Maxim Kontsevich's formality morphism.The main result of the fifth chapter is that the deformation of polyvector fields constructed in the fourth chapter is (generically) nontrivial and, in a sense, the unique such deformation. The proof is based on some cohomology computations involving Kontsevich's graph complex and related complexes. The chapter ends with an application of the results to properties of a derived version of the Duflo isomorphism.The sixth chapter develops a general mathematical framework for how and when an algebraic structure on the germs at the origin of a sheaf on Cartesian space can be "globalized" to a corresponding algebraic structure on the global sections over an arbitrary smooth manifold. The results are applied to the construction of the fourth chapter, and it is shown that the construction globalizes to polyvector fields and polydifferential operators on an arbitrary smooth manifold.The seventh chapter combines the relations to graph complexes, explained in chapter five, and the globalization theory of chapter six, to uncover a representation of the Grothendieck-Teichmüller group in terms of A-infinity morphisms between Poisson cohomology cochain complexes on a manifold.Chapter eight gives a simplified version of a construction of a family of Drinfel'd associators due to Carlo Rossi and Thomas Willwacher. Our simplified construction makes the connections to multiple zeta values more transparent--in particular, one obtains a fairly explicit family of evaluations on the algebra of formal multiple zeta values, and the chapter proves certain basic properties of this family of evaluations.
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Bode, Benjamin. "Knotted fields and real algebraic links." Thesis, University of Bristol, 2018. http://hdl.handle.net/1983/8527a201-2fba-4e7e-8481-3df228051413.
This doctoral thesis offers a new approach to the construction of arbitrarily knotted configurations in physical systems. We describe an algorithm that given any link $L$ finds a function $f:mathbb{C}^2tomathbb{C}$, a polynomial in complex variables $u$, $v$ and the conjugate $overline{v}$, whose vanishing set $f^{-1}(0)$ intersects the unit three-sphere $S^3$ in $L$. These functions can often be manipulated to satisfy the physical constraints of the system in question. The explicit construction allows us to make precise statements about properties of these functions, such as the polynomial degree and the number of critical points of $arg f$. Furthermore, we prove that for any link $L$ in an infinite family, namely the closures of squares of homogeneous braids, the polynomials can be altered into polynomials from $mathbb{R}^4$ to $mathbb{R}^2$ with an isolated singularity at the origin and $L$ as the link of that singularity. Links for which such polynomials exist are called real algebraic links and our explicit construction is a step towards their classification. We also study the crossing numbers of composite knots and relate them to crossing numbers of spatial graphs. The resulting connections are expected to lead to a new approach to the conjecture of the additivity of the crossing number.
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McCoy, Daisy Cox. "Irreducible elements in algebraic number fields." Diss., Virginia Tech, 1990. http://hdl.handle.net/10919/39950.
Alnaser, Ala' Jamil. "Waring's problem in algebraic number fields." Diss., Manhattan, Kan. : Kansas State University, 2009. http://hdl.handle.net/2097/2207.
NATO Advanced Study Institute on Higher-Dimensional Geometry over Finite Fields (2007 University of Göttingen). Higher-dimensional geometry over finite fields. Amsterdam, Netherlands: IOS Press, 2008.
Cohn, P. M. "Global fields." In Algebraic Numbers and Algebraic Functions, 83–108. Boston, MA: Springer US, 1991. http://dx.doi.org/10.1007/978-1-4899-3444-4_3.
Cohn, P. M. "Function fields." In Algebraic Numbers and Algebraic Functions, 109–75. Boston, MA: Springer US, 1991. http://dx.doi.org/10.1007/978-1-4899-3444-4_4.
Cohn, P. M. "Fields with valuations." In Algebraic Numbers and Algebraic Functions, 1–42. Boston, MA: Springer US, 1991. http://dx.doi.org/10.1007/978-1-4899-3444-4_1.
Bochnak, Jacek, Michel Coste, and Marie-Françoise Roy. "Ordered Fields, Real Closed Fields." In Real Algebraic Geometry, 7–21. Berlin, Heidelberg: Springer Berlin Heidelberg, 1998. http://dx.doi.org/10.1007/978-3-662-03718-8_2.
Pohst, Michael E. "Algebraic number fields." In Computational Algebraic Number Theory, 27–33. Basel: Birkhäuser Basel, 1993. http://dx.doi.org/10.1007/978-3-0348-8589-8_4.
Bordellès, Olivier. "Algebraic Number Fields." In Universitext, 517–673. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-54946-6_7.
Conference papers on the topic "Algebraic fields":
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Boku, Dereje Kifle, Wolfram Decker, Claus Fieker, and Andreas Steenpass. "Gröbner bases over algebraic number fields." In PASCO '15: International Workshop on Parallel Symbolic Computation. New York, NY, USA: ACM, 2015. http://dx.doi.org/10.1145/2790282.2790284.
Valasek, Gabor, and Robert Ban. "Higher Order Algebraic Signed Distance Fields." In CAD'22. CAD Solutions LLC, 2022. http://dx.doi.org/10.14733/cadconfp.2022.287-291.
Voight, John. "Curves over finite fields with many points: an introduction." In Computational Aspects of Algebraic Curves. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812701640_0010.
Efrat, Ido. "Recovering higher global and local fields from Galois groups – an algebraic approach." In Higher local fields. Mathematical Sciences Publishers, 2000. http://dx.doi.org/10.2140/gtm.2000.3.273.
Kapranov, Mikhail. "Harmonic analysis on algebraic groups over two-dimensional local fields of equal characteristic." In Higher local fields. Mathematical Sciences Publishers, 2000. http://dx.doi.org/10.2140/gtm.2000.3.255.
Tamura, Jun-ichi, Shin-ichi Yasutomi, and Takao Komatsu. "Algebraic Jacobi-Perron algorithm for biquadratic numbers." In DIOPHANTINE ANALYSIS AND RELATED FIELDS—2010: DARF—2010. AIP, 2010. http://dx.doi.org/10.1063/1.3478174.
Hajime, Kaneko, and Takao Komatsu. "Expansion of real numbers by algebraic numbers." In DIOPHANTINE ANALYSIS AND RELATED FIELDS: DARF 2007/2008. AIP, 2008. http://dx.doi.org/10.1063/1.2841897.
Huang, Yu-Chih. "Lattice index codes from algebraic number fields." In 2015 IEEE International Symposium on Information Theory (ISIT). IEEE, 2015. http://dx.doi.org/10.1109/isit.2015.7282903.
Tanaka, Taka-aki, Masaaki Amou, and Masanori Katsurada. "Algebraic independence properties related to certain infinite products." In DIOPHANTINE ANALYSIS AND RELATED FIELDS 2011: DARF - 2011. AIP, 2011. http://dx.doi.org/10.1063/1.3630047.
Encarnación, Mark J. "Factoring polynomials over algebraic number fields via norms." In the 1997 international symposium. New York, New York, USA: ACM Press, 1997. http://dx.doi.org/10.1145/258726.258802.
Xiao, M. DA (Differential Algebraic) Method and Symplectification for Field Map Generated Matrices of Siberian Snake. Office of Scientific and Technical Information (OSTI), September 1998. http://dx.doi.org/10.2172/1149860.
Schoen, Robert C., Daniel Anderson, and Charity Bauduin. Elementary Mathematics Student Assessment: Measuring Grade 3, 4, and 5 Students’ Performace in Number (Whole Numbers and Fractions), Operations, and Algebraic Thinking in Spring 2016. Florida State University Library, May 2018. http://dx.doi.org/10.33009/fsu.1653497279.
This report provides a description of the development process, field testing, and psychometric properties of a student mathematics test designed to assess grades 3, 4, and 5 student abilities. The test was administered to 2,754 participating grade 3, 4, and 5 students in 55 schools located in 10 public school districts in Florida during spring 2016. Focused on number (including whole number and fractions), operations, and algebraic thinking, the student assessment was designed to serve as a baseline measure of student achievement in a randomized controlled trial evaluating the impact of a teacher professional development program called Cognitively Guided Instruction (CGI) on student learning.
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Chang, P. A Differential Algebraic Integration Algorithm for Symplectic Mappings in Systems with Three-Dimensional Magnetic Field. Office of Scientific and Technical Information (OSTI), September 2004. http://dx.doi.org/10.2172/833057.
Bayak, Igor V. Applications of the Local Algebras of Vector Fields to the Modelling of Physical Phenomena. Jgsp, 2015. http://dx.doi.org/10.7546/jgsp-38-2015-1-23.
Caspi, S., M. Helm, L. J. Laslett, and V. O. Brady. An approach to 3D magnetic field calculation using numerical and differential algebra methods. Office of Scientific and Technical Information (OSTI), July 1992. http://dx.doi.org/10.2172/7252409.
