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Lectures on Algebra Weizhe Zheng September 30, 2021 Morningside Center of Mathematics and Hua Loo-Keng Key Laboratory of Mathematics Academy of Mathematics and Systems Science, Chinese Academy of Sciences Beijing 100190, China University of the Chinese Academy of Sciences, Beijing 100049, China Email: [email protected] Contents Preface v 1 Fields and Galois theory 1 1.1 Introduction: Solvability by radicals ................... 1 1.2 Fields and field extensions ........................ 3 1.3 Simple extensions ............................. 4 1.4 Algebraic extensions ........................... 6 1.5 Splitting fields and normal extensions .................. 8 1.6 Separable extensions ........................... 11 1.7 The Primitive Element Theorem ..................... 14 1.8 Finite Galois extensions ......................... 15 1.9 Irreducibility of polynomials ....................... 18 1.10 Galois groups of polynomials ....................... 20 1.11 Finite fields ................................ 23 1.12 Cyclotomic fields ............................. 23 1.13 Trace and norm .............................. 25 1.14 Hilbert’s Theorem 90 ........................... 27 1.15 Kummer and Artin–Schreier theories .................. 29 1.16 Solvability by radicals .......................... 33 1.17 Infinite Galois theory ........................... 34 1.18 Galois categories ............................. 39 1.19 Transcendental extensions ........................ 44 2 Modules 51 2.1 Modules and homomorphisms ...................... 51 2.2 Products and direct sums ........................ 52 2.3 Finitely generated modules over a PID ................. 55 2.4 Chain conditions ............................. 57 2.5 Semisimple modules ........................... 62 2.6 Indecomposable modules ......................... 64 2.7 Tensor products of modules ....................... 66 3 Rings and algebras 69 3.1 Algebras and tensor products ...................... 69 3.2 Semisimple rings ............................. 73 3.3 Simple rings ................................ 74 3.4 Jacobson radicals ............................. 75 iii iv CONTENTS 3.5 Semiprimitive and semiprimary rings .................. 77 3.6 Jacobson density theorem ........................ 78 3.7 Central simple algebras .......................... 81 3.8 Galois descent ............................... 88 4 Representations of finite groups 91 4.1 Group representations and modules ................... 91 4.2 Absolutely simple modules ........................ 94 4.3 Characters of finite groups ........................ 99 4.4 Induced representations ......................... 104 4.5 Representations of the symmetric group ................ 110 4.6 Induction theorems ............................ 116 4.7 Tannaka duality .............................. 120 Bibliography 123 Preface These lecture notes were prepared for the graduate course Algebra I from September 2019 to January 2020 at the University of the Chinese Academy of Sciences. We have followed more or less closely notes of Wen-Wei Li [L3] and Yongquan Hu [H2], who taught the course in previous years. I am deeply indebted to them for sharing their notes. I thank Zhijian Chen, Zhibin Geng, Peng Shan, and Hu Tan for corrections and suggestions. v vi PREFACE Chapter 1 Fields and Galois theory Convention 1.0.1. Rings are assumed to have identities and ring homomorphisms are assumed to preserve the identities. Some authors, notably Noether, do not follow this convention. 1.1 Introduction: Solvability by radicals Let F be a field. One goal of this chapter is to address the question of solvability by n radicals of a polynomial equation anX +···+a0 = 0 in one variable with coefficients in F . We let F [X] denote the ring of polynomials in X with coefficients in F . Example 1.1.1. Assume that the characteristic of F =6 2. Then the roots of a quadratic polynomial P (X) = aX2 + bX + c ∈ F [X] (a =6 0) are √ −b ± ∆ , 2a where ∆ = b2 − 4ac is the discriminant. Solutions of cubic and quartic equations by radicals were published by Cardano in 1545 and attributed to del Ferro and Tartaglia for cubic equations and Ferrari for quartic equations. Example 1.1.2. Assume that the characteristic of F =6 2, 3. Let P (X) = aX3 + bX2 + cX + d ∈ F [X] (a =6 0) be a cubic polynomial. Up to scaling we may assume − b 3 a = 1. Up to replacing X by X 3 , we may assume√P (X) = X + cX + d. Write − − − 1 − − P (X) = (X α1)(X α2)(X α3). Let ω = 2 ( 1 + 3) be a primitive cube root of unity. Consider the Lagrange resolvents 0 = α1 + α2 + α3, 2 β1 = α1 + ωα2 + ω α3, 2 β2 = α1 + ω α2 + ωα3 and the polynomial − − − 2 − − − 2 3− 3 3− 3 3 (Y β1)(Y ωβ1)(Y ω β1)(Y β2)(Y ωβ2)(Y ω β2) = (Y β1 )(Y β2 ) = Q(Y ). 1 2 CHAPTER 1. FIELDS AND GALOIS THEORY We have 3 3 2 · 2 · − β1 + β2 = (β1 + β2)(β1 + ωβ2)(β1 + ω β2) = 3α1 3ω α3 3ωα2 = 27d, 2 2 2 − − − − β1β2 = α1 + α2 + α3 α1α2 α2α3 α3α1 = 3c. 3 3 2 − 3 Thus β1 and β2 are the roots of the polynomial Q(Z) = Z + 27dZ 27c . Let γi = βi/3, i = 1, 2. Then (up to reordering) √ √ d d2 c3 d d2 c3 γ3 = − + + , γ3 = − − + . 1 2 4 27 2 2 4 27 Take (γ1, γ2) satisfying γ1γ2 = −c/3. Finally, 2 2 α1 = γ1 + γ2, α2 = ω γ1 + ωγ2, α3 = ωγ1 + ω γ2. Example 1.1.3. Assume that the characteristic of F =6 2, 3. Without loss of gen- erality, let P (X) = X4 + cX2 + dX + e ∈ F [X] be a quartic polynomial. Write P (X) = (X − α1)(X − α2)(X − α3)(X − α4). Consider 1 β = (α + α − α − α ), 1 2 1 2 3 4 1 β = (α − α + α − α ), 2 2 1 2 3 4 1 β = (α − α − α + α ) 3 2 1 2 3 4 and the polynomial − − − 2− 2 2− 2 2− 2 2 (Y β1)(Y +β1)(Y β2)(Y +β2)(Y β3)(Y +β3) = (Y β1 )(Y β2 )(Y β3 ) = Q(Y ). An elementary computation gives Q(Z) = Z3 + 2cZ2 + (c2 − 4e)Z − d2 ∈ F [Z]. One can thus find β1, β2, β3 by radicals up to signs. Finally 1 α = (β + β + β ), α = β − α , i = 1, 2, 3. 1 2 1 2 3 i+1 i 1 The correct signs can be determined by trial. By contrast, polynomial equations of degree ≥ 5 are not solvable by radicals in general. n n−1 Theorem 1.1.4 (Abel–Ruffini 1824). Let P (X) = X + T1X + ··· + Tn ∈ F (T1,...,Tn)[X] be the generic monic polynomial of degree n ≥ 5. Here F (T1,...,Tn) is the fraction field of the polynomial ring F [T1,...,Tn]. Then P (X) = 0 is not solv- able by radicals. We will deduce the theorem from Galois theory and Kummer theory. 1.2. FIELDS AND FIELD EXTENSIONS 3 1.2 Fields and field extensions Definition 1.2.1. A field is a nonzero commutative ring whose nonzero elements are invertible. Lemma 1.2.2. (1) A commutative ring A is a field if and only if 0 is a maximal ideal A. (2) Let F be a field and R a nonzero ring. Then every ring homomorphism ι: F → R an injection. Proof. (1) Indeed, A is a field ⇔ A is nonzero and aA = A for every nonzero a ∈ A ⇔ A is nonzero and the only ideals are 0 and A ⇔ 0 is a maximal ideal of A. (2) The kernel of ι is an ideal of F not containing 1, and thus must be zero by (1). In particular every ring homomorphism ι: F → E of fields is an embedding. The embedding ι factorizes as F ' ι(F ) ⊆ E, where F ' ι(F ) is an isomorphism (of fields, namely of rings). Given an inclusion of fields F ⊆ E (preserving 1), K is called a subfield of E and E is called a (field) extension of K, and we adopt the notation E/K (not to be confused with quotient). We sometimes extend the terminology of field extensions to field embeddings. Recall the characteristic of a ring R is the number n ∈ Z≥0 such that ker(Z → R) = nZ. The characteristic of a field is 0 or a prime number. A prime field is a field not containing any proper subfield. It is uniquely isomorphic to Q or Fp = Z/pZ for some prime number p. Every field F admits a smallest subfield of F , called the prime field of F . For any field embedding F,→ E, F and E have the same characteristic. Given a field extension E/F and a subset S ⊆ E, there exists a smallest subring F [S] ⊆ E containing F and S. This is a commutative domain and its fraction field F (S) ⊆ E is the smallest subfield containing F and S, called the subfield obtained by adjoining S to F or the subfield generated by S over F . Given field extensions K/F and K/F 0, the smallest subfield E ⊆ K containing F and F 0 is called the composite of F and F 0 in K and denoted by F · F 0 (or FF 0). Every element of F · F 0 has the form a b + ··· + a b 1 1 m m , c1d1 + ··· + cndn 0 where a1, . , am, c1, . , cn ∈ F and b1, . , bm, d1, . , dn ∈ F satisfying c1d1 + ··· + cndn =6 0. More generally, given a family of field extensions, (K/Fi)i∈I , the ⊆ smallest subfield∨E K containing the Fi’s is called the composite of (Fi)i∈I in K and denoted by i∈I Fi. We have ∨ ∪ ∨ Fi = Fj i∈I J⊆I j∈J where J runs through finite subsets of I. 4 CHAPTER 1. FIELDS AND GALOIS THEORY Warning 1.2.3. Given field embeddings ι: F,→ K, ι0 : F,→ K, the composite ι(F ) · ι0(F 0) depends on the choice of ι, ι0, and K. See Example 1.3.11 below. Given a field extension E/F , E is an F -vector space. Definition 1.2.4. The degree of a field extension E/F is [E : F ] = dimF (E) (a cardinal number). A field extension is said to be finite if [E : F ] is finite.
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