Abstract Algebra (Math 113) - Revision Sheet

1. 📌 Essentials

• Algebraic structures: sets with operations satisfying specific axioms (groups, rings, fields).
• Prime number: only divisible by 1 and itself.
• Group: set with an associative binary operation, identity, and inverses.
• Ring: set with addition and multiplication, distributive, with identity.
• Field: commutative ring with multiplicative invers for non-zero elements.
• Congruence mod m: a ≡ b mod m m | (a−b); forms residue classes.
• Cayley's theorem: any group G embeds into a symmetric group Σ(G).
• Normal subgroup: invariant under conjugation; allows quotient groups.
• Simple group: no non-trivial normal subgroups; building block of finite groups.
• Galois extension: normal + separable; automorphisms fixing base field.

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