## Basic Notions of Algebra (Google eBook)This book is wholeheartedly recommended to every student or user of mathematics. Although the author modestly describes his book as 'merely an attempt to talk about' algebra, he succeeds in writing an extremely original and highly informative essay on algebra and its place in modern mathematics and science. From the fields, commutative rings and groups studied in every university math course, through Lie groups and algebras to cohomology and category theory, the author shows how the origins of each algebraic concept can be related to attempts to model phenomena in physics or in other branches of mathematics. Comparable in style with Hermann Weyl's evergreen essay The Classical Groups, Shafarevich's new book is sure to become required reading for mathematicians, from beginners to experts. |

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### Contents

4 | |

10 Semisimple Modules and Rings | 79 |

ll Division Algebras of Finite Rank | 90 |

12 The Notion of a Group | 96 |

Finite Groups | 108 |

Inﬁnite Discrete Groups | 124 |

Lie Groups and Algebraic Groups | 140 |

16 General Results of Group Theory | 151 |

17 Group Representations | 160 |

Characters of compact Abelian groups and Fourier series Weyl and Ricci tensors in | 174 |

19 Lie Algebras and Nonassociative Algebra | 188 |

Categories | 202 |

21 Homological Algebra | 213 |

### Common terms and phrases

Abelian groups algebraic groups analogue arbitrary automorphism axioms called classiﬁcation coefﬁcients cohomology commutative ring compact complex analytic complex numbers consider consists construction contained coordinate coordinatisation corresponding coset curve deﬁned deﬁnition denoted dimension direct sum division algebra elements equation exact sequence Example exists extension L/K ﬁnd ﬁnite ﬁnite extension ﬁnite groups ﬁnite number ﬁnite rank ﬁnite type ﬁnite-dimensional ﬁrst ﬁxed follows functor g e G Galois geometry given GL(n group G Hence homomorphism identity inﬁnite integral invariant irreducible representations isomorphic kernel lattice Lie algebra Lie groups linear transformations manifold matrix module morphisms multiplication n-dimensional normal subgroup notion number ﬁeld permutations plane properties quaternions quotient rational functions real number reﬂection relations satisﬁes semisimple sheaf simple SL(n solvable submodule subspace symmetry group tensor Theorem theory topological space unique vector ﬁelds vector space