Courses for 2010/2011:

Math 2105 Introduction to Algebra, whole year

課程概述 | The Chinese remainder theorem. Euclid's algorithm. Rings, Isomorphisms. Groups, finite groups, homomorphism of groups, subgroups, normal subgroups, quotient groups, abelian groups. Direct products. Composition Series. Group actions. Classification of groups up to isomorphisms, Symmetric groups. Ideals. Ring homomorphisms. Unique factorization. Quotient rings. Commutative Rings. Matrix Rings. Field of fractions. Field extensions. Polynomials. Algebraic Equations. Geometric Constructions. Solvability by Radicals. Grobner basis. Vector Spaces over Fields. Finite Fields. Galois Theory, Fundamental Theorem of Algebra, Hilbert Nullstellen Satz. |

課程目標 | Introducing algebraic structures, algebra as a basic tool in Mathematics. Using the language of Algebra. Method of Algebra. Applications of Algebra. From Linear Algebra to Nonlinear Algebra. Classification of algebraic structures. |

關鍵字 | Groups, Rings, Fields, Vector Spaces, Homomorphisms, Isomorphisms, Polynomials, Algebraic Equations. |

課程要求 | High school Mathematics, Linear Algebra. |

指定閱讀 | J. Rotman: First Course in Abstract Algebra, 3rd ed. Prentice Hall 2005. |

參考書目 | M. Artin: Algebra, 2nd ed.
Prentice Hall, 1991. N. Jacobson: Basic Algebra 1, 2nd ed. W. Freeman, 1985. H. E. Rose: A course on Finite Groups, Springer UTM, 2009. D. Joyner: Adventures in Group Theory, Johns Hopkins U. Press, 2008. |

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