Ryota Matsuura, St. Olaf College, Northfield, MN.
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Description
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 $\mathbb{Z}_7[x]/ \langle x^2-1\rangle$ Quotient ring $\mathbb{R}[x]/ \langle x^2 +1\rangle$ $F[x]/ \langle g(x)\rangle$ 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