代數學(一)
Algebra (I)
Basic group theory and field theory, including finite fields and their applications in coding theory.
Linear Algebra
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| 週次 | 主題 |
|---|---|
| 第 1 週 | Introduction Sets Equivalence Relations |
| 第 2 週 | Roots of unity and $\Z_n$ Binary Operations Isomorphic Binary Structures Structure Properties |
| 第 3 週 | Groups Elementary Properties of Groups Group of Small Cardinalities Multiplicative group $\Z_n^ imes$ |
| 第 4 週 | Subgroups Cyclic Subgroups Cyclic Groups Direct Product |
| 第 5 週 | Groups of Permutation Symmetry Group of n-gons |
| 第 6 週 | Cayley Theorem Transpositions |
| 第 7 週 | Cosets Lagrange's Theorem Applications of Lagrange's Theorem |
| 第 8 週 | Homomorphisms Kernels of Homomorphisms |
| 第 9 週 | Midterm |
| 第 10 週 | Normal Subgroups Quotient Group Computation |
| 第 11 週 | Normal Subgroups II Classification of Finite Groups |
| 第 12 週 | Rings and Fields Ring Homomorphisms Integral Domains |
| 第 13 週 | Fields and Integral Domains Field of Quotients Ring of Polynomials |
| 第 14 週 | Division Algorithm of Polynomials Zeros of Polynomials Irreducible Polynomials |
| 第 15 週 | Polynomial Factorization Modulo a Prime Unique Factorization Theorem of F[x] Basis of Extension Fields |
| 第 16 週 | Algebraic Extensions Example of Algebraic Extensions Algebraic Closures |
| 第 17 週 | Existence and Uniqueness of Finite Fields Cyclotomic Polynomial over Finite Fields |
| 第 18 週 | Final Exam |
A First Course in Abstract Algebra, 7th Edition John B. Fraleigh
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