What are you going to learn?
Content
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Define a group, give examples of groups, list properties that hold in every group and state definitions of particular features of groups
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Define a ring, give examples of rings, list properties that hold in every ring and state definitions of particular features of rings
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Define a field, give examples of fields, list properties that hold in every field and state definitions of particular features of fields
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Prove statements about these mathematical structures
Chapter 1. Introduction
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Logic, Sets, Operations
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Equivalence Relations
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Equivalence Classes
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Permutations Groups
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Subgroups
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Groups and Symmetry
Chapter 2. Groups 1
Chapter 3. Groups 2
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Properties
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Generators. Direct Groups
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Cosets
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Lagranges’s Theorem. Cyclic Groups
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Isomorphisms
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Cayle’s Theorem
Chapter 4. Groups Homomorphisms
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Homomorphisms and Kernels
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Quotient Groups
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The Fundamental Homomorphism Theorem
Chapter 5. Introduction to Rings
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Definition and Examples
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Integral Domains, Division Rings
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Fields
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Isomorphisms. Characteristics
Chapter 6. Numbers Systems
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Ordered Integral Domains
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Integers
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Field of Quotients. Field of Rationals
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Ordered Fields. The Field of Reals
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The Field of Complex Numbers
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Complex Roots of Unity
Chapter 7. Polynomials
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Definitions and Elementary properties
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The Division Algorithm
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Factorization of Polynomials
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Unique Factorization Domains
Chapter 8. Quotient Rings
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Homomorphisms of Rings. Ideals
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Quotient Rings
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Factorization and Ideals
Chapter 9. Galois Theory
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Simple Extensions. Degree
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Roots of Polynomials
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Fundamental Theorem
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Algebraic Extensions
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Splitting Fields. Galois Groups
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Separability and Normality
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Fundamental Theorem of Galois Theory
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Solvability by Radicals
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Finite Fields
Bibliography
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Durbin, J.R. A Modern Algebra:An Introduction. 6th Ed. John Wiley and Sons, 2009.
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Rotman, J.J. A First Course in Abstract Algebra. 7th Ed. Prentice Hall
