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Algebraic Structures Algebraic Structures Groningen, 2nd year bachelor mathematics, 2017 (mostly a translation from original Dutch lecture notes of L.N.M. van Geemen, H.W. Lenstra, F. Oort, and J. Top). J. Top i Contents I Rings ..............................................................................2 I.1 Definition, examples, elementary properties . 2 I.2 Units and zero divisors. .5 I.3 Constructions of rings. .9 I.4 Exercises. .14 II Ring homomorphisms and ideals.............................................18 II.1 Ring homomorphisms. .18 II.2 Ideals . 19 II.3 The factor ring R/I............................................................22 II.4 Calculating with ideals . 26 II.5 Exercises. .31 III Rings of polynomials ..........................................................35 III.1 Polynomials . 35 III.2 Evaluation homomorphisms . 38 III.3 Division with remainder for polynomials . 40 III.4 Rings of polynomials over a field. .44 III.5 Rings of polynomials over a domain . 45 III.6 Differentiation . 46 III.7 Exercises. .49 IV Prime ideals and maximal ideals .............................................52 IV.1 Prime ideals . 52 IV.2 Maximal ideals . 54 IV.3 Zorn’s lemma . 55 IV.4 Exercises. .59 V Division in rings................................................................62 V.1 Irreducible elements . 62 V.2 Principal ideal domains. .63 V.3 Unique factorization domains . 65 V.4 Polynomials over unique factorization domains . 68 V.5 Factorizing and irreducibility of polynomials . 72 V.6 Exercises. .75 VI Euclidean rings and the Gaussian integers .................................78 VI.1 Euclidean rings . 78 VI.2 The Euclidean algorithm . 81 VI.3 Sums of squares . 83 VI.4 Exercises. .89 ii CONTENTS VII Fields ............................................................................91 VII.1 Prime fields and characteristic . 91 VII.2 Algebraic and transcendental. .92 VII.3 Finite and algebraic extensions. .95 VII.4 Determining a minimal polynomial. .98 VII.5 Exercises . 100 VIII Automorphisms of fields and splitting fields...............................102 VIII.1 Homomorphisms of fields. .102 VIII.2 Splitting fields . 105 VIII.3 Exercises . 110 IX Finite fields ....................................................................111 IX.1 Classification of finite fields.. .111 IX.2 The structure of finite fields . 113 IX.3 Irreducible polynomials over finite fields . 116 IX.4 The multiplicative group of a finite field. .119 IX.5 Exercises . 121 CONTENTS iii preface These lecture notes are based on a translation into English of the Dutch lecture notes Algebra II (Algebraic Structures) as they were used in the mathematics cur- riculum of Groningen University during the period 1993–2013. The original Dutch text may be found at http://www.math.rug.nl/~top/dic.pdf. Both the present text and the original build upon an earlier Dutch text on Rings and Fields, called Algebra II, written in the late 1970’s at the university of Amster- dam by Prof.dr. F. Oort and Prof.dr. H.W. Lenstra. In the 80’s L.N.M. van Geemen at Utrecht university added some chapters to their text, and in the 90’s in Gronin- gen I included various changes. The translation project consists of two parts. The first one (Algebraic Struc- tures) deals with the chapters 1 5, 7 9, and 12 of the Dutch notes. Many cor- ¡ ¡ rections and suggestions for improvement were offered by Dr. Max Kronberg who taught a course following these notes in the spring of 2017, and by Petra Hogeboom and Manoy Trip who at that time were students in this course. In the spring of 2019 student Wout Moltmaker mentioned a number of further small issues in the text, which have now been corrected. I am very grateful to all of them; needless to say that any mistakes and unclear parts in the exposition are only my fault. The second part of the translation project (which provides the material for the course Advanced Algebraic Structures and is not included here) discusses the chapters 8 (in slightly more detail than originally, to facilitate treating Galois theory), and chapters 6,13 on (projective) modules, then a discussion of basic Galois theory not present in the original notes, and finally the chapters 11 and 10 as well as an ex- tended version of chapter 14 and a brief discussion of tensor products. Groningen, January 2017 – April 2019 Jaap Top CONTENTS 1 I RINGS I.1 Definition, examples, elementary properties I.1.1 Definition. A ring (with 1) (also called unitary ring) is a five tuple (R, , ,0,1) Å ¢ with R a set, and maps written as: Å ¢ : R R R, (a, b) a b : R R R, (a, b) ab, Å £ ! 7! Å ¢ £ ! 7! and 0 and 1 elements of R, such that the following properties (R1) to (R4) hold: (R1) (R, ,0) is an abelian group; this means: Å (G1) a (b c) (a b) c for all a, b, c R; Å Å Æ Å Å 2 (G2) 0 a a 0 a for all a R; Å Æ Å Æ 2 (G3) every a R has an ‘opposite’ a R satisfying 2 ¡ 2 a ( a) ( a) a 0; Å ¡ Æ ¡ Å Æ (G4) a b b a for all a, b R. Å Æ Å 2 (R2) a(bc) (ab)c for all a, b, c R (associativity of ); Æ 2 ¢ (R3) a(b c) (ab) (ac) and (b c)a (ba) (ca) for all a, b, c R (the distributive Å Æ Å Å Æ Å 2 laws). (R4) 1a a1 a for all a R. Æ Æ 2 A ring R is called commutative if moreover (R5) holds: (R5) ab ba for all a, b R. Æ 2 If a, b R then a b and ab are called the sum and the product of a and b; the 2 Å product ab is also denoted as a b. The maps and are called the addition and ¢ Å ¢ the multiplication in R. If (R, , ,0,1) is a ring then one says that R is a ring with Å ¢ addition , multiplication , zero element 0, and unit element 1. Å ¢ A trivial example of a ring is the zero ring ({0}, , ,0,0), with 0 0 0 0 0. Å ¢ Å Æ ¢ Æ This is the only ring having 1 0. Æ Some textbooks define ‘rings’ (R, , ,0) only satisfying (R1), (R2), and (R3); such Å ¢ ‘rings’ are called non-unitary rings. A division ring (or skew field) is a ring R such that in addition to (R1) to (R4), also (R6) holds: 1 (R6) 1 0, and for all a R,a 0 there exists an inverse a¡ R satisfying 6Æ1 1 2 6Æ 2 a a¡ a¡ a 1. ¢ Æ ¢ Æ 2 I RINGS A field (French: corps; German: Körper, Dutch: lichaam, Flemish: veld) is a commutative division ring (so (R1) to (R6) hold). A simple example of a field is the set {0,1} with addition as in the abelian group Z/2Z and product 0 0 0 1 1 0 0 ¢ Æ ¢ Æ ¢ Æ and 1 1 1. The unit element is 1 ( 0), this field we denote by F2. ¢ Æ 6Æ I.1.2 Example. The sets Z, Q, R, C of integers and rational, real, complex numbers (respectively) are with the familiar addition and multiplication rings. Moreover Z,Q,R, and C are commutative. The rings Q,R, and C are fields, and Z is not a field (condition (R6) is not satisfied in Z). I.1.3 Example. Fix n Z 0. The set Z/nZ {0,1,..., n 1} with i i nZ Z, is 2 È Æ ¡ Æ Å ½ equipped with an addition, since the elements i are the residue classes with respect to the normal subgroup nZ Z. The rule ½ a b : a b, ¢ Æ ¢ with a b the familiar multiplication in Z defines a product (verify for yourself that ¢ this is well-defined: if a a1 and b b1, then indeed a b a1 b1). Æ Æ ¢ Æ ¢ With respect to this addition and multiplication Z/nZ is a commutative ring with unit element 1. In I.2.11 we will see that Z/nZ is a field if and only if n is a prime number. For n 1 we have Z/1Z which is the zero ring (hence, it is not a Æ field). I.1.4 Example. Let n Z 0..
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