Honors Abstract Algebra - Harvard University
Honors Abstract Algebra
Course Notes
Math 55a, Harvard University
Contents
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Introduction . . . . . . . . . . . .
Set Theory . . . . . . . . . . . . .
Vector spaces . . . . . . . . . . .
Polynomials . . . . . . . . . . . .
Linear Operators . . . . . . . . .
Inner product spaces . . . . . . .
Bilinear forms . . . . . . . . . . .
Trace and determinant . . . . . .
Introduction to Group Theory . .
Symmetry . . . . . . . . . . . . .
Finite group theory . . . . . . . .
Representation theory . . . . . .
Group presentations . . . . . . .
Knots and the fundamental group
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Introduction
This course will provide a rigorous introduction to abstract algebra, including
group theory and linear algebra.
Topics include:
1. Set theory. Formalization of Z, Q, R, C.
2. Linear algebra. Vector spaces and transformations over R and C. Other
ground fields. Eigenvectors. Jordan form.
3. Multilinear algebra. Inner products, quadratic forms, alternating forms,
tensor products, determinants.
4. Abstract groups.
5. Groups, symmetry and representations.
1
2
Set Theory
Halmos reading. Read Halmos, Naive Set Theory, sections 1C15 to learn
the foundations of mathematics from the point of view of set theory, and
its use in formalizing the integers. Most of this should be review, although
the systematic use of a small set of axioms to rigorously establish set theory
may be new to you. We will also formalize the rational, real and complex
numbers.
Then read 22-23 to learn about cardinality and countable sets.
Finally, read 16C21 and 24C25 to learn about other versions of the axiom
of choice, ordinals and cardinals.
Axiom of choice. For any set A, there is a function c : P(A) ? {?} A
such that c(B) B for all B ? A.
Theorem 2.1 The Axiom of Choice is equivalent to the assertion that every
set can be well-ordered.
Proof. If (A, ................
................
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