• The integers

• Preview of groups by examples
• Basic group properties
• Group actions: centralizer, normalizer as isotropy
• Cyclic groups
• Kernels and quotients
• Cosets in Lagrange's Theorem and in actions
• Simplicity of the alternating group
• The three group isomorphism theorems
• Abelian quotients
• The group product via its mapping property
• Finitely-generated abelian groups
• The Sylow theorems
• Classification of small groups

• Euclidean rings
• Ideals of a commutative ring, quotient field of an integral domain
• Algebraic structure and generative differentiation rules | (shorter version)

• What is a polynomial?
• Factorization of polynomials
• Symmetric polynomials
• Cyclotomic polynomials

• Free modules
• Zorn's Lemma (writeup never used, hence untested)
• Modules over a PID
• Canonical forms in linear algebra

• Multilinear algebra: the tensor product
• Multilinear algebra: the exterior product
• The Cayley-Hamilton theorem via multilinear algebra

• The seventeenth root of unity via quadratics
• Cyclotomic-intermediate fields via Gauss sums

• Assignment 1
• Assignment 2
• Assignment 3
• Assignment 4
• Assignment 5

• Quiz | A useful little fact for the quiz

• Homework problem on rings and ideals
• Homework problems on symmetric polynomials
• Homework problem on modules over a PID
• More homework problems on modules over a PID
• Homework problem on hom-group formation as an adjoint functor | Template
• Homework problems on the Cayley-Hamilton argument

Comments and corrections should be sent to jerry@reed.edu.

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