Mor04, Mor12], Provides a Foundational Tool for 1 Solving Problems in A1-Enumerative Geometry
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Lecture Notes in Galois Theory
Lecture Notes in Galois Theory Lectures by Dr Sheng-Chi Liu Throughout these notes, signifies end proof, and N signifies end of example. Table of Contents Table of Contents i Lecture 1 Review of Group Theory 1 1.1 Groups . 1 1.2 Structure of cyclic groups . 2 1.3 Permutation groups . 2 1.4 Finitely generated abelian groups . 3 1.5 Group actions . 3 Lecture 2 Group Actions and Sylow Theorems 5 2.1 p-Groups . 5 2.2 Sylow theorems . 6 Lecture 3 Review of Ring Theory 7 3.1 Rings . 7 3.2 Solutions to algebraic equations . 10 Lecture 4 Field Extensions 12 4.1 Algebraic and transcendental numbers . 12 4.2 Algebraic extensions . 13 Lecture 5 Algebraic Field Extensions 14 5.1 Minimal polynomials . 14 5.2 Composites of fields . 16 5.3 Algebraic closure . 16 Lecture 6 Algebraic Closure 17 6.1 Existence of algebraic closure . 17 Lecture 7 Field Embeddings 19 7.1 Uniqueness of algebraic closure . 19 Lecture 8 Splitting Fields 22 8.1 Lifts are not unique . 22 Notes by Jakob Streipel. Last updated December 6, 2019. i TABLE OF CONTENTS ii Lecture 9 Normal Extensions 23 9.1 Splitting fields and normal extensions . 23 Lecture 10 Separable Extension 26 10.1 Separable degree . 26 Lecture 11 Simple Extensions 26 11.1 Separable extensions . 26 11.2 Simple extensions . 29 Lecture 12 Simple Extensions, continued 30 12.1 Primitive element theorem, continued . 30 Lecture 13 Normal and Separable Closures 30 13.1 Normal closure . 31 13.2 Separable closure . 31 13.3 Finite fields . -
MRD Codes: Constructions and Connections
MRD Codes: Constructions and Connections John Sheekey April 12, 2019 This preprint is of a chapter to appear in Combinatorics and finite fields: Difference sets, polynomials, pseudorandomness and applications. Radon Series on Computational and Applied Mathematics, K.-U. Schmidt and A. Winterhof (eds.). The tables on clas- sifications will be periodically updated (in blue) when further results arise. If you have any data that you would like to share, please contact the author. Abstract Rank-metric codes are codes consisting of matrices with entries in a finite field, with the distance between two matrices being the rank of their difference. Codes with maximum size for a fixed minimum distance are called Maximum Rank Distance (MRD) codes. Such codes were constructed and studied independently by Delsarte (1978), Gabidulin (1985), Roth (1991), and Cooperstein (1998). Rank-metric codes have seen renewed interest in recent years due to their applications in random linear network coding. MRD codes also have interesting connections to other topics such as semifields (finite nonassociative division algebras), finite geometry, linearized polynomials, and cryptography. In this chapter we will survey the known constructions and applications of MRD codes, and present some open problems. arXiv:1904.05813v1 [math.CO] 11 Apr 2019 1 Definitions and Preliminaries 1.1 Rank-metric codes Coding theory is the branch of mathematics concerned with the efficient and accurate transfer of information. Error-correcting codes are used when communication is over a channel in which errors may occur. This requires a set equipped with a distance function, and a subset of allowed codewords; if errors are assumed to be small, then a received message is decoded to the nearest valid codeword. -
Counting and Effective Rigidity in Algebra and Geometry
COUNTING AND EFFECTIVE RIGIDITY IN ALGEBRA AND GEOMETRY BENJAMIN LINOWITZ, D. B. MCREYNOLDS, PAUL POLLACK, AND LOLA THOMPSON ABSTRACT. The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2–manifolds (resp., 3–manifolds). By work of Reid, this spectrum determines the commensurability class of the 2–manifold (resp., 3–manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length that can be explicitly bounded as a function of volume. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3–manifolds. These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest. 1. INTRODUCTION 1.1. Inverse problems. 1.1.1. Algebraic problems. Given a degree d central division algebra D over a field k, the set of isomorphism classes of maximal subfields MF(D) of D is a basic and well studied invariant of D. Question 1. Do there exist non-isomorphic, central division algebras D1;D2=k with MF(D1) = MF(D2)? Restricting to the class of number fields k, by a well-known consequence of class field theory, when D=k is a quaternion algebra, MF(D) = MF(D0) if and only if D =∼ D0 as k–algebras. -
ADJOINT DIVISORS and FREE DIVISORS Contents 1. Introduction
ADJOINT DIVISORS AND FREE DIVISORS DAVID MOND AND MATHIAS SCHULZE Abstract. We describe two situations where adding the adjoint divisor to a divisor D with smooth normalization yields a free divisor. Both also involve stability or versality. In the first, D is the image of a corank 1 stable map-germ (Cn, 0) → (Cn+1, 0), and is not free. In the second, D is the discriminant of a versal deformation of a weighted homogeneous function with isolated critical point (subject to certain numerical conditions on the weights). Here D itself is already free. We also prove an elementary result, inspired by these first two, from which we obtain a plethora of new examples of free divisors. The presented results seem to scratch the surface of a more general phenomenon that is still to be revealed. Contents 1. Introduction 1 Notation 4 Acknowledgments 4 2. Images of stable maps 4 3. Discriminants of hypersurface singularities 12 4. Pull-back of free divisors 20 References 22 1. Introduction Let M be an n-dimensional complex analytic manifold and D a hypersurface in M. The OM -module Der( log D) of logarithmic vector fields along D consists of all vector fields on M tangent− to D at all smooth points of D. If this module is locally free, D is called a free divisor. This terminology was introduced by Kyoji Saito in [Sai80b]. As freeness is evidently a local condition, so we may pass to n germs of analytic spaces D X := (C , 0), pick coordinates x1,...,xn on X and a defining equation h O ⊂= C x ,...,x for D. -
Arxiv:2007.12573V3 [Math.AC] 28 Aug 2020 1,Tm .] Egtamr Rcs Eso Fteegnau Theore Eigenvalue the of Version Precise 1.2 More a Theorem Get We 3.3], Thm
STICKELBERGER AND THE EIGENVALUE THEOREM DAVID A. COX To David Eisenbud on the occasion of his 75th birthday. Abstract. This paper explores the relation between the Eigenvalue Theorem and the work of Ludwig Stickelberger (1850-1936). 1. Introduction The Eigenvalue Theorem is a standard result in computational algebraic geom- etry. Given a field F and polynomials f1,...,fs F [x1,...,xn], it is well known that the system ∈ (1.1) f = = fs =0 1 ··· has finitely many solutions over the algebraic closure F of F if and only if A = F [x ,...,xn]/ f ,...,fs 1 h 1 i has finite dimension over F (see, for example, Theorem 6 of [7, Ch. 5, 3]). § A polynomial f F [x ,...,xn] gives a multiplication map ∈ 1 mf : A A. −→ A basic version of the Eigenvalue Theorem goes as follows: Theorem 1.1 (Eigenvalue Theorem). When dimF A< , the eigenvalues of mf ∞ are the values of f at the finitely many solutions of (1.1) over F . For A = A F F , we have a canonical isomorphism of F -algebras ⊗ A = Aa, ∈V a F Y(f1,...,fs) where Aa is the localization of A at the maximal ideal corresponding to a. Following [12, Thm. 3.3], we get a more precise version of the Eigenvalue Theorem: arXiv:2007.12573v3 [math.AC] 28 Aug 2020 Theorem 1.2 (Stickelberger’s Theorem). For every a V (f ,...,fs), we have ∈ F 1 mf (Aa) Aa, and the restriction of mf to Aa has only one eigenvalue f(a). ⊆ This result easily implies Theorem 1.1 and enables us to compute the character- istic polynomial of mf . -
Computing Isomorphisms and Embeddings of Finite
Computing isomorphisms and embeddings of finite fields Ludovic Brieulle, Luca De Feo, Javad Doliskani, Jean-Pierre Flori and Eric´ Schost May 4, 2017 Abstract Let Fq be a finite field. Given two irreducible polynomials f; g over Fq, with deg f dividing deg g, the finite field embedding problem asks to compute an explicit descrip- tion of a field embedding of Fq[X]=f(X) into Fq[Y ]=g(Y ). When deg f = deg g, this is also known as the isomorphism problem. This problem, a special instance of polynomial factorization, plays a central role in computer algebra software. We review previous algorithms, due to Lenstra, Allombert, Rains, and Narayanan, and propose improvements and generalizations. Our detailed complexity analysis shows that our newly proposed variants are at least as efficient as previously known algorithms, and in many cases significantly better. We also implement most of the presented algorithms, compare them with the state of the art computer algebra software, and make the code available as open source. Our experiments show that our new variants consistently outperform available software. Contents 1 Introduction2 2 Preliminaries4 2.1 Fundamental algorithms and complexity . .4 2.2 The Embedding Description problem . 11 arXiv:1705.01221v1 [cs.SC] 3 May 2017 3 Kummer-type algorithms 12 3.1 Allombert's algorithm . 13 3.2 The Artin{Schreier case . 18 3.3 High-degree prime powers . 20 4 Rains' algorithm 21 4.1 Uniquely defined orbits from Gaussian periods . 22 4.2 Rains' cyclotomic algorithm . 23 5 Elliptic Rains' algorithm 24 5.1 Uniquely defined orbits from elliptic periods . -
Hidden Number Problems
Hidden Number Problems Barak Shani A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics The University of Auckland 2017 Abstract The hidden number problem is the problem of recovering an unknown group element (the \hidden number") given evaluations of some function on products of the hidden number with known elements in the group. This problem enjoys a vast variety of applications, and provides cross-fertilisation among different areas of mathematics. Bit security is a research field in mathematical cryptology that studies leakage of in- formation in cryptographic systems. Of particular interest are public-key cryptosystems, where the study revolves around the information about the private keys that the public keys leak. Ideally no information is leaked, or more precisely extraction of partial in- formation about the secret keys is (polynomially) equivalent to extraction of the entire keys. Accordingly, studies in this field focus on reducing the problem of recovering the private key to the problem of recovering some information about it. This is done by designing algorithms that use the partial information to extract the keys. The hidden number problem was originated to study reductions of this kind. This thesis studies the hidden number problem in different groups, where the functions are taken to output partial information on the binary representation of the input. A spe- cial focus is directed towards the bit security of Diffie–Hellman key exchange. The study presented here provides new results on the hardness of extracting partial information about Diffie–Hellman keys. Contents 1 Introduction 1 1.1 Summary of Contributions . -
Constructive and Computational Aspects of Cryptographic Pairings
Constructive and Computational Aspects of Cryptographic Pairings Michael Naehrig Constructive and Computational Aspects of Cryptographic Pairings PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr.ir. C.J. van Duijn, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op donderdag 7 mei 2009 om 16.00 uur door Michael Naehrig geboren te Stolberg (Rhld.), Duitsland Dit proefschrift is goedgekeurd door de promotor: prof.dr. T. Lange CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN Naehrig, Michael Constructive and Computational Aspects of Cryptographic Pairings / door Michael Naehrig. – Eindhoven: Technische Universiteit Eindhoven, 2009 Proefschrift. – ISBN 978-90-386-1731-2 NUR 919 Subject heading: Cryptology 2000 Mathematics Subject Classification: 94A60, 11G20, 14H45, 14H52, 14Q05 Printed by Printservice Technische Universiteit Eindhoven Cover design by Verspaget & Bruinink, Nuenen c Copyright 2009 by Michael Naehrig Fur¨ Lukas und Julius Promotor: prof.dr. T. Lange Commissie: prof.dr.dr.h.c. G. Frey (Universit¨at Duisburg-Essen) prof.dr. M. Scott (Dublin City University) prof.dr.ir. H.C.A. van Tilborg prof.dr. A. Blokhuis prof.dr. D.J. Bernstein (University of Illinois at Chicago) prof.dr. P.S.L.M. Barreto (Universidade de S˜ao Paulo) Alles, was man tun muss, ist, die richtige Taste zum richtigen Zeitpunkt zu treffen. Johann Sebastian Bach Thanks This dissertation would not exist without the help, encouragement, motivation, and company of many people. I owe much to my supervisor, Tanja Lange. I thank her for her support; for all the efforts she made, even in those years, when I was not her PhD student; for taking care of so many things; and for being a really good supervisor. -
DERIVATIONS PRESERVING a MONOMIAL IDEAL 1. Introduction
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 137, Number 9, September 2009, Pages 2935–2942 S 0002-9939(09)09922-5 Article electronically published on May 4, 2009 DERIVATIONS PRESERVING A MONOMIAL IDEAL YOHANNES TADESSE (Communicated by Bernd Ulrich) Abstract. Let I be a monomial ideal in a polynomial ring A = k[x1,...,xn] over a field k of characteristic 0, TA/k(I) be the module of I-preserving k- derivations on A and G be the n-dimensional algebraic torus on k. We compute the weight spaces of TA/k(I) considered as a representation of G.Usingthis, we show that TA/k(I) preserves the integral closure of I and the multiplier ideals of I. 1. Introduction Throughout this paper TA/k denotes the module of k-linear derivations of the polynomial ring A = k[x1,...,xn]overafieldk of characteristic 0. Let G = k∗ ×···×k∗ be the n-dimensional algebraic torus on k. There is an action of θ θ1 ··· θn · θ θ1 ··· θn · G on a monomial X = x1 xn defined by (t1,...,tn) X =(t1 tn ) Xθ, and we say that Xθ has weight θ.ThismakesA arepresentationofG and induces an action of G on TA/k which will make TA/k arepresentationsuchthat θ ∈ − ⊆ X ∂xj TA/k has weight (θ1,...,θj 1,...θn). Let I A be a monomial ideal, i.e. a G-invariant ideal. Let TA/k(I)={δ ∈ TA/k | δ(I) ⊆ I}⊂TA/k be the submodule of I-preserving derivations, which is a G-subrepresentation of TA/k. Our first result is a description of the weight spaces of TA/k(I). -
Tagungsbericht 18/1996 Kommutative Algebra Und Algebraische
Tagungsbericht 18/1996 Kommutative Algebra und algebraische Geometrie 05.05. - 11.05.1996 Die Tagung fand unter der Leitung von Ernst Kunz (Regensburg), Josepf Lipman (West La.fayette (USA» und Uwe Storch (Bochum) statt. In den Vorträgen und Diskussionen wurde über neuere Ergebnisse und Probleme aus der kommutativen Algebra und algebraischen Geometrie berichtet und es wurden insbesondere Fragen angesprochen, die heiden Gebieten gemeinsam ent springen. Auch Verbindungen zu anderen Bereichen der Mathematik (Homotopie theorie, K -Theorie, Kombinatorik) wurden geknüpft. Eine Reihe von Vorträ.gen beschäftigte sich" mit Residuen und Spuren in der Dualitätstheorie. Weitere Schwer punktsthemen waren Fundamentalgruppen algebraischer Varietäten, birationale Geometrie, Frobenius- F\mktoren, Auflösungen und Hilbertfunktionen graduier ter Ringe und Moduln, insbesondere in Verbindung mit der Geometrie endlicher Punktmengen im pn und kombinatorischen Strukturen. In diesem Zusammen hang sind auch mehrere Vorträge über die· MultipllZität lokaler und graduierter Ringe zu sehen. Das große Interesse an der Tagung spiegelt sich auch in der hohen Zahl ausländischer Gäste wider. Neben den 19 deutsChen Teilnehmern kamen 10 weitere aus eu ropäischen Ländern (davon 3 aus Osteuropa), 12 aus Nordamerika und je einer aus Indien, Israel, Japan und Taiwan. / 1 /- © • ~'''' ~ ~ ... ~ ': I ~~ .., .., : ;" .. ' --~~ \.... ... ;' - Vortragsauszüge Abhyankar, Shreeram.S. Local Fundamental Gronps of Algebraic Varieties For a. prime number p and integer t ~ 0, let Q,(p) = the set of all quasi (p, t) group8, i. e., finite groups G such that G/p{G) is generated by t generators, where p(G) is the subgroup of G generated by all of its p-Sylow sub~oups; members of. Qo(p) are called quasi-p groUp8. -
Arxiv:2010.09374V1 [Math.AG] 19 Oct 2020 Lsdfils N Eoescasclcut Over Counts Classical Recovers One fields
Applications to A1-enumerative geometry of the A1-degree Sabrina Pauli Kirsten Wickelgren Abstract These are lecture notes from the conference Arithmetic Topology at the Pacific Institute of Mathemat- ical Sciences on applications of Morel’s A1-degree to questions in enumerative geometry. Additionally, we give a new dynamic interpretation of the A1-Milnor number inspired by the first named author’s enrichment of dynamic intersection numbers. 1 Introduction A1-homotopy theory provides a powerful framework to apply tools from algebraic topology to schemes. In these notes, we discuss Morel’s A1-degree, giving the analog of the Brouwer degree in classical topology, and applications to enumerative geometry. Instead of the integers, the A1-degree takes values in bilinear forms, or more precisely, in the Grothendieck-Witt ring GW(k) of a field k, defined to be the group completion of isomorphism classes of symmetric, non-degenerate bilinear k-forms. This can result in an enumeration of algebro-geometric objects valued in GW(k), giving an A1-enumerative geometry over non-algebraically closed fields. One recovers classical counts over C using the rank homomorphism GW(k) Z, but GW(k) can contain more information. This information can record arithmetic-geometric properties→ of the objects being enumerated over field extensions of k. In more detail, we start with the classical Brouwer degree. We introduce enough A1-homotopy theory to describe Morel’s degree and use the Eisenbud-Khimshiashvili-Levine signature formula to give context for the degree and a formula for the local A1-degree. The latter is from joint work of Jesse Kass and the second-named author. -
LECTURES on COHOMOLOGICAL CLASS FIELD THEORY Number
LECTURES on COHOMOLOGICAL CLASS FIELD THEORY Number Theory II | 18.786 | Spring 2016 Taught by Sam Raskin at MIT Oron Propp Last updated August 21, 2017 Contents Preface......................................................................v Lecture 1. Introduction . .1 Lecture 2. Hilbert Symbols . .6 Lecture 3. Norm Groups with Tame Ramification . 10 Lecture 4. gcft and Quadratic Reciprocity. 14 Lecture 5. Non-Degeneracy of the Adèle Pairing and Exact Sequences. 19 Lecture 6. Exact Sequences and Tate Cohomology . 24 Lecture 7. Chain Complexes and Herbrand Quotients . 29 Lecture 8. Tate Cohomology and Inverse Limits . 34 Lecture 9. Hilbert’s Theorem 90 and Cochain Complexes . 38 Lecture 10. Homotopy, Quasi-Isomorphism, and Coinvariants . 42 Lecture 11. The Mapping Complex and Projective Resolutions . 46 Lecture 12. Derived Functors and Explicit Projective Resolutions . 52 Lecture 13. Homotopy Coinvariants, Abelianization, and Tate Cohomology. 57 Lecture 14. Tate Cohomology and Kunr ..................................... 62 Lecture 15. The Vanishing Theorem Implies Cohomological lcft ........... 66 Lecture 16. Vanishing of Tate Cohomology Groups. 70 Lecture 17. Proof of the Vanishing Theorem . 73 Lecture 18. Norm Groups, Kummer Theory, and Profinite Cohomology . 76 Lecture 19. Brauer Groups . 81 Lecture 20. Proof of the First Inequality . 86 Lecture 21. Artin and Brauer Reciprocity, Part I. 92 Lecture 22. Artin and Brauer Reciprocity, Part II . 96 Lecture 23. Proof of the Second Inequality . 101 iii iv CONTENTS Index........................................................................ 108 Index of Notation . 110 Bibliography . 113 Preface These notes are for the course Number Theory II (18.786), taught at mit in the spring semester of 2016 by Sam Raskin. The original course page can be found online here1; in addition to these notes, it includes an annotated bibliography for the course, as well as problem sets, which are frequently referenced throughout the notes.