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Field Theory Pete L. Clark
Field Theory Pete L. Clark Thanks to Asvin Gothandaraman and David Krumm for pointing out errors in these notes. Contents About These Notes 7 Some Conventions 9 Chapter 1. Introduction to Fields 11 Chapter 2. Some Examples of Fields 13 1. Examples From Undergraduate Mathematics 13 2. Fields of Fractions 14 3. Fields of Functions 17 4. Completion 18 Chapter 3. Field Extensions 23 1. Introduction 23 2. Some Impossible Constructions 26 3. Subfields of Algebraic Numbers 27 4. Distinguished Classes 29 Chapter 4. Normal Extensions 31 1. Algebraically closed fields 31 2. Existence of algebraic closures 32 3. The Magic Mapping Theorem 35 4. Conjugates 36 5. Splitting Fields 37 6. Normal Extensions 37 7. The Extension Theorem 40 8. Isaacs' Theorem 40 Chapter 5. Separable Algebraic Extensions 41 1. Separable Polynomials 41 2. Separable Algebraic Field Extensions 44 3. Purely Inseparable Extensions 46 4. Structural Results on Algebraic Extensions 47 Chapter 6. Norms, Traces and Discriminants 51 1. Dedekind's Lemma on Linear Independence of Characters 51 2. The Characteristic Polynomial, the Trace and the Norm 51 3. The Trace Form and the Discriminant 54 Chapter 7. The Primitive Element Theorem 57 1. The Alon-Tarsi Lemma 57 2. The Primitive Element Theorem and its Corollary 57 3 4 CONTENTS Chapter 8. Galois Extensions 61 1. Introduction 61 2. Finite Galois Extensions 63 3. An Abstract Galois Correspondence 65 4. The Finite Galois Correspondence 68 5. The Normal Basis Theorem 70 6. Hilbert's Theorem 90 72 7. Infinite Algebraic Galois Theory 74 8. A Characterization of Normal Extensions 75 Chapter 9. -
K-Quasiderivations
K-QUASIDERIVATIONS CALEB EMMONS, MIKE KREBS, AND ANTHONY SHAHEEN Abstract. A K-quasiderivation is a map which satisfies both the Product Rule and the Chain Rule. In this paper, we discuss sev- eral interesting families of K-quasiderivations. We first classify all K-quasiderivations on the ring of polynomials in one variable over an arbitrary commutative ring R with unity, thereby extend- ing a previous result. In particular, we show that any such K- quasiderivation must be linear over R. We then discuss two previ- ously undiscovered collections of (mostly) nonlinear K-quasiderivations on the set of functions defined on some subset of a field. Over the reals, our constructions yield a one-parameter family of K- quasiderivations which includes the ordinary derivative as a special case. 1. Introduction In the middle half of the twientieth century|perhaps as a reflection of the mathematical zeitgeist|Lausch, Menger, M¨uller,N¨obauerand others formulated a general axiomatic framework for the concept of the derivative. Their starting point was (usually) a composition ring, by which is meant a commutative ring R with an additional operation ◦ subject to the restrictions (f + g) ◦ h = (f ◦ h) + (g ◦ h), (f · g) ◦ h = (f ◦ h) · (g ◦ h), and (f ◦ g) ◦ h = f ◦ (g ◦ h) for all f; g; h 2 R. (See [1].) In M¨uller'sparlance [9], a K-derivation is a map D from a composition ring to itself such that D satisfies Additivity: D(f + g) = D(f) + D(g) (1) Product Rule: D(f · g) = f · D(g) + g · D(f) (2) Chain Rule D(f ◦ g) = [(D(f)) ◦ g] · D(g) (3) 2000 Mathematics Subject Classification. -
Algorithmic Factorization of Polynomials Over Number Fields
Rose-Hulman Institute of Technology Rose-Hulman Scholar Mathematical Sciences Technical Reports (MSTR) Mathematics 5-18-2017 Algorithmic Factorization of Polynomials over Number Fields Christian Schulz Rose-Hulman Institute of Technology Follow this and additional works at: https://scholar.rose-hulman.edu/math_mstr Part of the Number Theory Commons, and the Theory and Algorithms Commons Recommended Citation Schulz, Christian, "Algorithmic Factorization of Polynomials over Number Fields" (2017). Mathematical Sciences Technical Reports (MSTR). 163. https://scholar.rose-hulman.edu/math_mstr/163 This Dissertation is brought to you for free and open access by the Mathematics at Rose-Hulman Scholar. It has been accepted for inclusion in Mathematical Sciences Technical Reports (MSTR) by an authorized administrator of Rose-Hulman Scholar. For more information, please contact [email protected]. Algorithmic Factorization of Polynomials over Number Fields Christian Schulz May 18, 2017 Abstract The problem of exact polynomial factorization, in other words expressing a poly- nomial as a product of irreducible polynomials over some field, has applications in algebraic number theory. Although some algorithms for factorization over algebraic number fields are known, few are taught such general algorithms, as their use is mainly as part of the code of various computer algebra systems. This thesis provides a summary of one such algorithm, which the author has also fully implemented at https://github.com/Whirligig231/number-field-factorization, along with an analysis of the runtime of this algorithm. Let k be the product of the degrees of the adjoined elements used to form the algebraic number field in question, let s be the sum of the squares of these degrees, and let d be the degree of the polynomial to be factored; then the runtime of this algorithm is found to be O(d4sk2 + 2dd3). -
Section 1.2 – Mathematical Models: a Catalog of Essential Functions
Section 1-2 © Sandra Nite Math 131 Lecture Notes Section 1.2 – Mathematical Models: A Catalog of Essential Functions A mathematical model is a mathematical description of a real-world situation. Often the model is a function rule or equation of some type. Modeling Process Real-world Formulate Test/Check problem Real-world Mathematical predictions model Interpret Mathematical Solve conclusions Linear Models Characteristics: • The graph is a line. • In the form y = f(x) = mx + b, m is the slope of the line, and b is the y-intercept. • The rate of change (slope) is constant. • When the independent variable ( x) in a table of values is sequential (same differences), the dependent variable has successive differences that are the same. • The linear parent function is f(x) = x, with D = ℜ = (-∞, ∞) and R = ℜ = (-∞, ∞). • The direct variation function is a linear function with b = 0 (goes through the origin). • In a direct variation function, it can be said that f(x) varies directly with x, or f(x) is directly proportional to x. Example: See pp. 26-28 of the text. 1 Section 1-2 © Sandra Nite Polynomial Functions = n + n−1 +⋅⋅⋅+ 2 + + A function P is called a polynomial if P(x) an x an−1 x a2 x a1 x a0 where n is a nonnegative integer and a0 , a1 , a2 ,..., an are constants called the coefficients of the polynomial. If the leading coefficient an ≠ 0, then the degree of the polynomial is n. Characteristics: • The domain is the set of all real numbers D = (-∞, ∞). • If the degree is odd, the range R = (-∞, ∞). -
On the Discriminant of a Certain Quartinomial and Its Totally Complexness
ON THE DISCRIMINANT OF A CERTAIN QUARTINOMIAL AND ITS TOTALLY COMPLEXNESS SHUICHI OTAKE AND TONY SHASKA Abstract. In this paper, we compute the discriminant of a quartinomial n 2 of the form f(b;a;1)(t; x) = x + t(x + ax + b) by using the Bezoutian. Then, by using this result and another theorem of our previous paper, we construct a family of totally complex polynomials of the form f(b;a;1)(ξ; x) (ξ 2 R; (a; b) 6= (0; 0)). 1. Introduction The discriminant of a polynomial has been a major objective of research in algebra because of its importance. For example, the discriminant of a polynomial allows us to know whether the polynomial has multiple roots or not and it also plays an important role when we compute the discriminant of a number field. Moreover the discriminant of a polynomial also tells us whether the Galois group of the polynomial is contained in the alternating group or not. This is why there are so many papers focusing on studying the discriminant of a polynomial ([B-B-G], [D-S], [Ked], [G-D], [Swa]). In the last two papers, the authors concern the computation of the discriminant of a trinomial and it has been carried out in different ways. n In this paper, we compute the discriminant of a quartinomial f(b;a;1)(t; x) = x + t(x2 + ax + b) by using the Bezoutian (Theorem2). Let F be a field of characteristic zero and f1(x), f2(x) be polynomials over F . Then, for any integer n such that n ≥ maxfdegf1; degf2g, we put n f1(x)f2(y) − f1(y)f2(x) X B (f ; f ) : = = α xn−iyn−j 2 F [x; y]; n 1 2 x − y ij i;j=1 Mn(f1; f2) : = (αij)1≤i;j≤n: 0 The n × n matrix Mn(f1; f2) is called the Bezoutian of f1 and f2. -
Solvability by Radicals Rodney Coleman, Laurent Zwald
Solvability by radicals Rodney Coleman, Laurent Zwald To cite this version: Rodney Coleman, Laurent Zwald. Solvability by radicals. 2020. hal-03066602 HAL Id: hal-03066602 https://hal.archives-ouvertes.fr/hal-03066602 Preprint submitted on 15 Dec 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Solvability by radicals Rodney Coleman, Laurent Zwald December 10, 2020 Abstract In this note we present one of the fundamental theorems of algebra, namely Galois’s theorem concerning the solution of polynomial equations. We will begin with a study of cyclic extensions, before moving onto radical extensions. 1 Cyclic extensions We say that a finite extension E of a field F is cyclic if the Galois group Gal(E=F ) is cyclic. In this section we aim to present some elementary properties of such extensions. We begin with a preliminary result known as Hilbert’s theorem 90. EXTSOLVcycth1 Theorem 1 Let E=F be a finite cyclic Galois extension of degree n. We suppose that σ is a generator of the Galois group Gal(E=F ). If α 2 E, then NE=F (α) = 1 if and only if there exists ∗ β β 2 E such that α = σ(β) . -
FIELD EXTENSION REVIEW SHEET 1. Polynomials and Roots Suppose
FIELD EXTENSION REVIEW SHEET MATH 435 SPRING 2011 1. Polynomials and roots Suppose that k is a field. Then for any element x (possibly in some field extension, possibly an indeterminate), we use • k[x] to denote the smallest ring containing both k and x. • k(x) to denote the smallest field containing both k and x. Given finitely many elements, x1; : : : ; xn, we can also construct k[x1; : : : ; xn] or k(x1; : : : ; xn) analogously. Likewise, we can perform similar constructions for infinite collections of ele- ments (which we denote similarly). Notice that sometimes Q[x] = Q(x) depending on what x is. For example: Exercise 1.1. Prove that Q[i] = Q(i). Now, suppose K is a field, and p(x) 2 K[x] is an irreducible polynomial. Then p(x) is also prime (since K[x] is a PID) and so K[x]=hp(x)i is automatically an integral domain. Exercise 1.2. Prove that K[x]=hp(x)i is a field by proving that hp(x)i is maximal (use the fact that K[x] is a PID). Definition 1.3. An extension field of k is another field K such that k ⊆ K. Given an irreducible p(x) 2 K[x] we view K[x]=hp(x)i as an extension field of k. In particular, one always has an injection k ! K[x]=hp(x)i which sends a 7! a + hp(x)i. We then identify k with its image in K[x]=hp(x)i. Exercise 1.4. Suppose that k ⊆ E is a field extension and α 2 E is a root of an irreducible polynomial p(x) 2 k[x]. -
DISCRIMINANTS in TOWERS Let a Be a Dedekind Domain with Fraction
DISCRIMINANTS IN TOWERS JOSEPH RABINOFF Let A be a Dedekind domain with fraction field F, let K=F be a finite separable ex- tension field, and let B be the integral closure of A in K. In this note, we will define the discriminant ideal B=A and the relative ideal norm NB=A(b). The goal is to prove the formula D [L:K] C=A = NB=A C=B B=A , D D ·D where C is the integral closure of B in a finite separable extension field L=K. See Theo- rem 6.1. The main tool we will use is localizations, and in some sense the main purpose of this note is to demonstrate the utility of localizations in algebraic number theory via the discriminants in towers formula. Our treatment is self-contained in that it only uses results from Samuel’s Algebraic Theory of Numbers, cited as [Samuel]. Remark. All finite extensions of a perfect field are separable, so one can replace “Let K=F be a separable extension” by “suppose F is perfect” here and throughout. Note that Samuel generally assumes the base has characteristic zero when it suffices to assume that an extension is separable. We will use the more general fact, while quoting [Samuel] for the proof. 1. Notation and review. Here we fix some notations and recall some facts proved in [Samuel]. Let K=F be a finite field extension of degree n, and let x1,..., xn K. We define 2 n D x1,..., xn det TrK=F xi x j . -
Trigonometric Functions
72 Chapter 4 Trigonometric Functions To define the radian measurement system, we consider the unit circle in the xy-plane: ........................ ....... ....... ...... ....................... .............. ............... ......... ......... ....... ....... ....... ...... ...... ...... ..... ..... ..... ..... ..... ..... .... ..... ..... .... .... .... .... ... (cos x, sin x) ... ... 4 ... A ..... .. ... ....... ... ... ....... ... .. ....... .. .. ....... .. .. ....... .. .. ....... .. .. ....... .. .. ....... ...... ....... ....... ...... ....... x . ....... Trigonometric Functions . ...... ....y . ....... (1, 0) . ....... ....... .. ...... .. .. ....... .. .. ....... .. .. ....... .. .. ....... .. ... ...... ... ... ....... ... ... .......... ... ... ... ... .... B... .... .... ..... ..... ..... ..... ..... ..... ..... ..... ...... ...... ...... ...... ....... ....... ........ ........ .......... .......... ................................................................................... An angle, x, at the center of the circle is associated with an arc of the circle which is said to subtend the angle. In the figure, this arc is the portion of the circle from point (1, 0) So far we have used only algebraic functions as examples when finding derivatives, that is, to point A. The length of this arc is the radian measure of the angle x; the fact that the functions that can be built up by the usual algebraic operations of addition, subtraction, radian measure is an actual geometric length is largely responsible for the usefulness of -
Coefficients of Algebraic Functions: Formulae and Asymptotics
COEFFICIENTS OF ALGEBRAIC FUNCTIONS: FORMULAE AND ASYMPTOTICS CYRIL BANDERIER AND MICHAEL DRMOTA Abstract. This paper studies the coefficients of algebraic functions. First, we recall the too-less-known fact that these coefficients fn always a closed form. Then, we study their asymptotics, known to be of the type n α fn ∼ CA n . When the function is a power series associated to a context-free grammar, we solve a folklore conjecture: the appearing critical exponents α belong to a subset of dyadic numbers, and we initiate the study the set of possible values for A. We extend what Philippe Flajolet called the Drmota{Lalley{Woods theorem (which is assuring α = −3=2 as soon as a "dependency graph" associated to the algebraic system defining the function is strongly connected): We fully characterize the possible singular behaviors in the non-strongly connected case. As a corollary, it shows that certain lattice paths and planar maps can not be generated by a context-free grammar (i.e., their generating function is not N-algebraic). We give examples of Gaussian limit laws (beyond the case of the Drmota{Lalley{Woods theorem), and examples of non Gaussian limit laws. We then extend our work to systems involving non-polynomial entire functions (non-strongly connected systems, fixed points of entire function with positive coefficients). We end by discussing few algorithmic aspects. Resum´ e.´ Cet article a pour h´erosles coefficients des fonctions alg´ebriques.Apr`esavoir rappel´ele fait trop peu n α connu que ces coefficients fn admettent toujours une forme close, nous ´etudionsleur asymptotique fn ∼ CA n . -
Complex Algebraic Geometry
Complex Algebraic Geometry Jean Gallier∗ and Stephen S. Shatz∗∗ ∗Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] ∗∗Department of Mathematics University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] February 25, 2011 2 Contents 1 Complex Algebraic Varieties; Elementary Theory 7 1.1 What is Geometry & What is Complex Algebraic Geometry? . .......... 7 1.2 LocalStructureofComplexVarieties. ............ 14 1.3 LocalStructureofComplexVarieties,II . ............. 28 1.4 Elementary Global Theory of Varieties . ........... 42 2 Cohomologyof(Mostly)ConstantSheavesandHodgeTheory 73 2.1 RealandComplex .................................... ...... 73 2.2 Cohomology,deRham,Dolbeault. ......... 78 2.3 Hodge I, Analytic Preliminaries . ........ 89 2.4 Hodge II, Globalization & Proof of Hodge’s Theorem . ............ 107 2.5 HodgeIII,TheK¨ahlerCase . .......... 131 2.6 Hodge IV: Lefschetz Decomposition & the Hard Lefschetz Theorem............... 147 2.7 ExtensionsofResultstoVectorBundles . ............ 162 3 The Hirzebruch-Riemann-Roch Theorem 165 3.1 Line Bundles, Vector Bundles, Divisors . ........... 165 3.2 ChernClassesandSegreClasses . .......... 179 3.3 The L-GenusandtheToddGenus .............................. 215 3.4 CobordismandtheSignatureTheorem. ........... 227 3.5 The Hirzebruch–Riemann–Roch Theorem (HRR) . ............ 232 3 4 CONTENTS Preface This manuscript is based on lectures given by Steve Shatz for the course Math 622/623–Complex Algebraic Geometry, during Fall 2003 and Spring 2004. The process for producing this manuscript was the following: I (Jean Gallier) took notes and transcribed them in LATEX at the end of every week. A week later or so, Steve reviewed these notes and made changes and corrections. After the course was over, Steve wrote up additional material that I transcribed into LATEX. The following manuscript is thus unfinished and should be considered as work in progress. -
Uwe Krey · Anthony Owen Basic Theoretical Physics Uwe Krey · Anthony Owen
Uwe Krey · Anthony Owen Basic Theoretical Physics Uwe Krey · Anthony Owen Basic Theoretical Physics AConciseOverview With 31 Figures 123 Prof. Dr. Uwe Krey University of Regensburg (retired) FB Physik Universitätsstraße 31 93053 Regensburg, Germany E-mail: [email protected] Dr. rer nat habil Anthony Owen University of Regensburg (retired) FB Physik Universitätsstraße 31 93053 Regensburg, Germany E-mail: [email protected] Library of Congress Control Number: 2007930646 ISBN 978-3-540-36804-5 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting and production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: eStudio Calamar S.L., F. Steinen-Broo, Pau/Girona, Spain Printed on acid-free paper SPIN 11492665 57/3180/YL - 5 4 3 2 1 0 Preface This textbook on theoretical physics (I-IV) is based on lectures held by one of the authors at the University of Regensburg in Germany.