§ Abstract Algebra

Topics

Group theory

• Algebraic groups
• Abelian groups
• Normal subgroups and quotient groups
• Group homomorphisms and isomorphisms
• Free products of groups
• Group generators, relations, and presentations

Theorems:

• (Theorem) Lagrange
• (Theorem) van Kampen

General ring theory

Field theory

Theorems:

• (Theorem) Every nonconstant polynomial over a field has a root in some extension field
• (Theorem) Universal property of simple field extensions
• (Theorem) The number of extensions of a field homomorphism is at most the degree of the field extension

Galois theory

• Galois groups

Theorems:

• (Theorem) Fundamental theorem of Galois theory

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References

• Notes on Modern Algebra I by Patrick Gallagher