# Abstract Algebra [AA-101A]

The course structure and material follow Harvard course on Abstract Algebra:

http://wayback.archive-it.org/3671/20150528171650/https://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

COURSE CONTENT:

Chapter-1-Matrices

• Row reduction, row echelon matrix and theorems, Inverse matrix, Determinants and theorems, Permutation, Symmetric group, Cyclic notation , transposition, permutation matrix, sign of permutation, Other formulas for determinants, Co-factor matrix.

Chapter-2-Groups

• Law of Composition, Groups and Sub-groups, Abelian Group, General Linear Group, Order of group, Infinite abelian group, cancellation law, permutation group, General Linear group, symmetric group, subgroup, Special linear group, Proper subgroup, Subgroup of the additive group of $\mathbb{Z}$, gcd and Euclidean algorithm [upto 18];

Here, you will find the content from “Contemporary Abstract Algebra” by J.A. Gallian (7th Edition).

Page for this Book. (Include Prerequisites, Chapters and its contents links, Solutions to Exercises links )

Chapter-5- Permutation Groups

$\; \bullet \text { Permutation (P) of A and Permutation group of A } \; \mid \bullet \text { Symmetric group} \; \mid \bullet \text{ Cycle Notation } \; \bullet \text { Permutation as product of disjoint cycles } \; \mid \bullet \text { Condition for P to commute } \; \mid \bullet \text { Order of a P } \; \mid \bullet \text{ P as product of 2-cycles } \; \mid \bullet \text { Even and Odd permutations} \; \mid \bullet \text { Alternating Group }$

Chapter-6- Isomorphisms

$\; \bullet \text { Group Isomorphim(I) } \; \mid \bullet \text { Cayley's theorem} \; \mid \bullet \text{ Properties of I acting on elements and groups } \; \bullet \text { Automorphism (A.) } \; \mid \bullet \text { Inner A. induced by a } \; \mid \bullet \text { Inner A. (Inn(G)) and A of G (Aut(G)) } \; \mid \bullet \text{ Isomorphism between } Aut(\mathbb{Z}_n) \approx U(n)$

Solutions to the problems in the book.

Chapter-10- Group Homomorphisms

$\; \bullet \text { Group Homomorphism (H) } \; \mid \bullet \text { Kernel of H } \; \mid \bullet \text{ Properties of elements under H } \; \mid \bullet \text { Properties of subgroups under H } \; \mid \bullet \text { Kernels of H are normal subgroups } \; \mid \bullet \text { First Isomorphism theorem } \; \mid \bullet \text{ N/C theorem } \; \mid \bullet \text { Normal subgroups as kernel of H of Group }$

Exercise Solution: [7]

Chapter-12- Introduction to Rings

$\; \bullet \text {Ring } \; \mid \bullet \text { Unity in Ring} \; \mid \bullet \text{ Unit of the ring } \; \mid \bullet \text { Subrings } \; \mid \bullet \text { Gaussian Integers } \; \mid \bullet \text { } \; \mid \bullet \text{ } \; \mid \bullet \text { }$

Exercise Solutions [1-13]

Chapter-14- Ideal and Factor Rings

$\; \bullet \text { Ideals and Ideal Test } \; \mid \bullet \text { Principal Ideal generated by a} \; \mid \bullet \text{ Factor ring } \; \mid \bullet \text { Existence of factor rings } \; \mid \bullet \text { Prime Ideals and Maximal Ideals } \\ \; \mid \bullet \text { R/A Is an Integral Domain Iff A Is Prime Ideal } \\ \; \mid \bullet \text{ R/A Is a Field If and Only If A Is Maximal}$

Chapter-15- Ring Homomorphisms

$\; \bullet \text { Ring Homo(RH) and Ring Isomor (RI) } \; \mid \bullet \text { Natural H from}Z \text { to } Z_{n} \; \mid \bullet \text{ Test for divisibility by 9 } \; \mid \bullet \text { Theorem of Gersonides } \; \mid \bullet \text { Properties of RH } \; \mid \bullet \text { Isomorphism theorem for Rings } \; \mid \bullet \text{ Natural Homomorphism from R to R/A } \; \mid \bullet \text { Field of Quotients }$

Notes

Exercise Solutions [1-10+] [12]

Chapter-16- Polynomial Rings

$\; \bullet \text { Ring of Polynomials over R } \; \mid \bullet \text { Degree of Polynomial } \; \mid \bullet \text{ Leading coefficient of Polynomial } \; \mid \bullet \text { Monic Polynomial } \; \mid \bullet \text { Division Algorithm for f(x) } \; \mid \bullet \text { factor of f(x) } \; \mid \bullet \text{ Zero of a polynomial } \; \mid \bullet \text { Zero of multiplicity K }\; \mid \bullet \text { The remainder theorem } \; \mid \bullet \text{ The factor theorem } \; \mid \bullet \text { Primitive n-th root of unity } \; \mid \bullet \text { Principal ideal domain }$

Exercise Solutions [1-18]

Chapter-17- Factorization of Polynomials

Notes and Solutions

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