A Friendly Introduction to Abstract Algebra

ISBN-13: 9781470468811 / Miękka / 2022 / 367 str.

Ryota Matsuura

Spis treści

• Preliminaries: Introduction to proofs
• Sets and subsets
• Divisors
• Examples of groups: Modular arithmetic
• Symmetries
• Permutations
• Matrices
• Introduction to groups: Introduction to groups
• Groups of small size
• Matrix groups
• Subgroups
• Order of an element
• Cyclic groups, Part I
• Cyclic groups, Part II
• Group homomorphisms: Functions
• Isomorphisms
• Homomorphisms, Part I
• Homomorphisms, Part II
• Quotient groups: Introduction to cosets
• Lagrange's theorem
• Multiplying/adding cosets
• Quotient group examples
• Quotient group proofs
• Normal subgroups
• First isomorphism theorem
• Introduction to rings: Introduction to rings
• Integral domains and fields
• Polynomial rings, Part I
• Polynomial rings, Part II
• Factoring polynomials
• Quotient rings: Ring homomorphisms
• Introduction to quotient rings
• Quotient ring $\mathbb_7[x]/ \langle x^2-1\rangle$
• Quotient ring $\mathbb[x]/ \langle x^2 +1\rangle$
• $F[x]/ \langle g(x)\rangle$ is/isn't a field, Part I
• Maximal ideals
• $F[x]/ \langle g(x)\rangle$ is/isn't a field, Part II
• Appendices: Proof of the GCD theorem
• Composition table for $D_4$
• Symbols and notations
• Essential theorems
• Index: Index of terms