Abstract Algebra

This course contains 44 visible knowledge chapters in nine content-sized units of lengths 4, 4, 5, 6, 5, 4, 5, 7, and 4. Clock arithmetic, rearranging objects, and rotating a square look different. Why do many of the same reasoning patterns work in all three? Begin with a small operation table and check closure, associativity, identity, inverse, and commutativity separately. Groups are then built through subgroups, cyclic structure, direct products, permutations, actions, conjugacy, the class equation required by Sylow theory, cosets, Cauchy and Sylow theorems, quotients, correspondence, and all three elementary isomorphism theorems. The second half develops rings, ideals, domains, fraction fields, quotient rings, the Chinese remainder theorem, the Euclidean-domain-to-PID-to-UFD ladder, polynomial Euclid and Bézout, irreducible-versus-prime boundaries, minimal polynomials, extension degrees, and the tower law. Every new noun is tied to a test the learner can perform, every quotient calculation checks representative independence, and the final projects classify the groups of order four and reconstruct D₄ and F₄ from complete structural evidence.

Before this course: Completed Linear Algebra and Proof, Logic & Set Theory. The course assumes functions, equivalence relations, quotient sets, induction, contradiction, countability, matrices, linear transformations, bases, and rigorous proof, but no number theory, real analysis, topology, category theory, or Galois theory.

Chapter 1

A binary operation is a closed typed function

Objective: Why prove representative independence? Give the deciding definition, computation, or theorem.

An operation on S must send every ordered pair in S×S back into S; familiar formulas can fail closure when the underlying set changes. The conclusion to justify is: Addition modulo n is a well-defined associative binary operation on residue classes ℤ/nℤ. The worked result to reconstruct is: The table is a Latin square with identity [0].

Start from the objects and operation: A binary operation * on S is a function *:S×S→S. The first boundary to test is: Subtraction on positive integers is not a binary operation because 2−5 is outside the set. A quick self-check is: Why prove representative independence?

Addition modulo n is a well-defined associative binary operation on residue classes ℤ/nℤ.

Proof step 1: If a≡a′ and b≡b′ mod n, then a+b≡a′+b′, so representatives give the same class.

Proof step 2: The result [a+b] is again a residue class, proving closure.

Proof step 3: Integer associativity gives ([a]+[b])+[c]=[a]+([b]+[c]).

Worked problem: Build the addition table for ℤ/3ℤ. State the structure and operation before calculating.

Worked step 1: Use classes [0],[1],[2].
Worked step 2: Add representatives and reduce modulo 3.
Worked step 3: Rows are 0,1,2; 1,2,0; and 2,0,1.

Result: The table is a Latin square with identity [0].