Editing Abstract Algebra (section)

== <span style="color: #FFFFFF;">Remembering</span> ==
* '''Algebraic structure''' — A set equipped with one or more operations satisfying specified axioms; the central object of abstract algebra.
* '''Group''' — A set G with an operation · satisfying: closure, associativity, identity (e·a = a·e = a), and inverses.
* '''Abelian group''' — A group where the operation is commutative (a·b = b·a); named after Niels Abel.
* '''Ring''' — An abelian group under addition with a distributive multiplication; e.g., integers ℤ, polynomials, matrices.
* '''Field''' — A ring where every non-zero element has a multiplicative inverse; e.g., ℚ, ℝ, ℂ, finite fields 𝔽_p.
* '''Homomorphism''' — A structure-preserving map between algebraic structures; f(a·b) = f(a)·f(b).
* '''Isomorphism''' — A bijective homomorphism; two isomorphic structures are "algebraically identical."
* '''Subgroup''' — A subset of a group that is itself a group under the same operation.
* '''Normal subgroup''' — A subgroup N of G where gNg⁻¹ = N for all g ∈ G; needed to form quotient groups.
* '''Quotient group (factor group)''' — G/N: the group of cosets of a normal subgroup N in G.
* '''Lagrange's Theorem''' — The order of any subgroup of a finite group divides the order of the group.
* '''Sylow Theorems''' — Theorems about the existence and structure of subgroups of prime-power order in finite groups.
* '''Galois theory''' — Studies field extensions and their automorphism groups; resolves which polynomial equations are solvable by radicals.
* '''Vector space''' — A set with addition and scalar multiplication over a field; the setting for linear algebra.
* '''Ideal''' — A subset of a ring absorbing multiplication; quotient rings are formed using ideals.
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