Master groups, rings, fields, and algebraic structures
← Back to MathematicsLearn the fundamental concept of groups and their basic properties.
Study symmetric groups and the fundamental role of permutations.
Explore the structure of groups through cosets and fundamental counting principles.
Study structure-preserving maps between groups and their properties.
Introduction to rings, integral domains, and their fundamental properties.
Study polynomial rings and their arithmetic properties.
Explore field extensions and algebraic structures of fields.
Connect field extensions with group theory through Galois theory.
Explore advanced topics in group theory and classification.
Explore connections to other areas of mathematics and advanced structures.
Build the foundation of abstract algebra with the fundamental concept of groups.
Understand binary operations as functions from S × S to S. Learn closure property and explore examples like addition, multiplication, and composition.
Master the formal definition of a group (G, ∗) as a set with a binary operation satisfying four axioms: closure, associativity, identity, and inverses.
Study each axiom in detail: associativity (a∗b)∗c = a∗(b∗c), identity element e, and inverse elements. Learn to verify these properties.
Explore concrete examples: (ℤ, +), (ℚ*, ·), matrix groups, symmetry groups. Understand how abstract concepts apply to familiar structures.
Study commutative groups where a∗b = b∗a for all elements. Learn that commutativity is not required for groups but creates important special cases.
Define the order of an element as the smallest positive integer n such that aⁿ = e. Understand finite and infinite order elements.
Learn when a subset H of group G forms a subgroup. Master the subgroup test: H is non-empty and closed under the operation and inverses.
Study groups generated by a single element. Understand that ⟨a⟩ = {aⁿ : n ∈ ℤ} and learn properties of finite and infinite cyclic groups.
Explore the symmetric groups and understand groups through permutations and symmetries.
Understand permutations as bijective functions from a set to itself. Learn function composition as the group operation for permutations.
Study Sₙ, the group of all permutations of n elements. Understand that |Sₙ| = n! and learn the fundamental role of symmetric groups.
Master cycle notation for writing permutations. Learn to decompose permutations into disjoint cycles and understand why disjoint cycles commute.
Study 2-cycles (transpositions) and learn that every permutation can be written as a product of transpositions. Understand even and odd permutations.
Learn about Aₙ, the group of even permutations. Understand that Aₙ is a normal subgroup of Sₙ with index 2 for n ≥ 2.
Study Dₙ, the symmetry group of regular n-gons. Learn the presentation Dₙ = ⟨r, s | rⁿ = s² = e, srs = r⁻¹⟩ and geometric interpretation.
Understand how groups can act on sets. Learn the connection between group actions and permutation representations of abstract groups.
Prove that every group is isomorphic to a subgroup of some symmetric group. Understand this fundamental result showing permutation groups' universality.
Discover the structure of groups through cosets and prove fundamental counting results.
Define left cosets aH = {ah : h ∈ H} and right cosets Ha. Learn that cosets partition the group and understand their geometric interpretation.
Understand that each coset can be represented by any of its elements. Learn to find complete sets of coset representatives.
Prove that |G| = |H| · [G:H] where [G:H] is the index of H in G. Understand this fundamental counting principle in group theory.
Apply Lagrange's theorem: order of elements divides group order, groups of prime order are cyclic, and Fermat's Little Theorem connections.
Study the index [G:H] as the number of left (or right) cosets of H in G. Learn properties of index and its multiplicative behavior in towers.
Define normal subgroups N ⊴ G where left and right cosets coincide: gN = Ng for all g ∈ G. Learn equivalent characterizations.
Construct quotient groups G/N when N is normal. Understand the group operation (aN)(bN) = (ab)N and verify it's well-defined.
Study groups with no non-trivial normal subgroups. Understand their role as "building blocks" and learn examples like prime cyclic groups.
Study structure-preserving maps between groups and the fundamental isomorphism theorems.
Learn that φ: G → H is a homomorphism if φ(ab) = φ(a)φ(b). Understand how homomorphisms preserve group structure.
Define ker(φ) = {g ∈ G : φ(g) = eH} and im(φ) = {φ(g) : g ∈ G}. Learn that kernels are normal subgroups and images are subgroups.
Prove that homomorphisms preserve identities, inverses, and powers. Learn that φ is injective iff ker(φ) = {e}.
Study bijective homomorphisms that preserve all group structure. Understand that isomorphic groups are "essentially the same" algebraically.
Prove G/ker(φ) ≅ im(φ) for any homomorphism φ: G → H. Understand this fundamental connection between quotients and images.
Learn the Diamond Isomorphism Theorem: for subgroups H, K with K normal in HK, we have HK/K ≅ H/(H ∩ K).
Study the correspondence between subgroups of G/N and subgroups of G containing N. Understand (G/N)/(H/N) ≅ G/H when N ⊆ H.
Study isomorphisms from a group to itself. Learn about inner automorphisms φₐ(x) = axa⁻¹ and the automorphism group Aut(G).
Introduction to rings, integral domains, and their fundamental properties.
Learn that a ring (R, +, ·) is a set with two operations where (R, +) is an abelian group and multiplication is associative with distributive laws.
Master the ring axioms: additive group structure, associativity of multiplication, and left/right distributive laws connecting the operations.
Study units (invertible elements) and zero divisors (non-zero elements whose product is zero). Understand that units and zero divisors are disjoint.
Learn that integral domains are commutative rings with unity and no zero divisors. Study the cancellation property in integral domains.
Understand fields as commutative rings where every non-zero element is a unit. Learn that fields are integral domains with additional structure.
Study when a subset forms a subring. Master the subring test: non-empty, closed under subtraction and multiplication.
Learn structure-preserving maps φ: R → S where φ(a + b) = φ(a) + φ(b) and φ(ab) = φ(a)φ(b). Study kernels and images.
Study ideals as special subsets that "absorb" multiplication from the ring. Learn that kernels of ring homomorphisms are ideals.
Study polynomial rings and their arithmetic properties in detail.
Construct polynomial rings as formal expressions with coefficients in R. Understand polynomial addition and multiplication operations.
Define degree of polynomials and leading coefficients. Learn how degrees behave under addition and multiplication of polynomials.
Master polynomial long division: for f, g ∈ F[x] with g ≠ 0, there exist unique q, r with f = gq + r and deg(r) < deg(g).
Extend gcd concepts to polynomials. Learn the Euclidean algorithm for polynomials and how to express gcd as linear combinations.
Study polynomials that cannot be factored into lower-degree polynomials. Learn irreducibility tests and examples over different fields.
Prove that polynomial rings over fields have unique factorization into irreducible factors. Understand the role of principal ideal domains.
Learn that a is a root of f(x) iff (x - a) divides f(x). Study the relationship between roots and factorization over fields.
Study special properties when the coefficient ring is a field. Learn that F[x] is a principal ideal domain and Euclidean domain.
Explore field extensions and algebraic structures of fields.
Study field extensions E/F where F ⊆ E are fields. Understand how E can be viewed as a vector space over F.
Learn about extensions F(α) generated by a single element. Understand the difference between F(α) and F[α] and when they coincide.
Study elements α where some polynomial with coefficients in F has α as a root. Learn the distinction between algebraic and transcendental elements.
For algebraic elements, study the unique monic irreducible polynomial having α as a root. Learn properties and applications of minimal polynomials.
Define [E:F] as the dimension of E as an F-vector space. Learn the multiplicativity of degrees: [E:F] = [E:K][K:F].
Study the smallest field containing F and all roots of a polynomial. Learn that splitting fields exist and are unique up to isomorphism.
Understand algebraically closed fields where every polynomial has a root. Learn about algebraic closures and their uniqueness properties.
Study fields with finitely many elements. Learn that finite fields have prime power order and understand their structure and uniqueness.
Connect field extensions with group theory through the beautiful theory of Galois.
Study extensions E/F that are both normal (splitting field of separable polynomials) and separable. These are the extensions where Galois theory applies.
Learn that Gal(E/F) is the group of field automorphisms of E that fix F pointwise. Study how this group encodes the extension's structure.
For a group G of automorphisms, study E^G = {α ∈ E : σ(α) = α for all σ ∈ G}. This creates a correspondence between groups and fields.
Master the bijective correspondence between subgroups of Gal(E/F) and intermediate fields of E/F, with inclusion order reversed.
Understand when polynomial equations can be solved using arithmetic operations and nth roots. Connect this to solvable groups.
Learn why general quintic equations cannot be solved by radicals. Understand how Galois theory provides the definitive answer to this classical problem.
Study polynomials whose roots are primitive nth roots of unity. Learn their irreducibility and connections to number theory.
Apply Galois theory to classical problems: which regular polygons are constructible, impossibility of trisecting angles and doubling cubes.
Explore advanced topics in group theory including Sylow theorems and group actions.
Master the three Sylow theorems about the existence, conjugacy, and counting of Sylow p-subgroups. These are fundamental tools for group analysis.
Study groups whose order is a prime power. Learn their special properties and role in the structure theory of finite groups.
Formalize group actions as homomorphisms G → Sym(X). Study orbits, stabilizers, and how actions reveal group structure.
Prove |G| = |Orb(x)| · |Stab(x)| for group actions. Apply this fundamental counting principle to various problems.
Learn to count orbits using |X/G| = (1/|G|) Σ_{g∈G} |X^g|. Apply this to combinatorial problems and symmetry counting.
Study groups with composition series where all factors are abelian. Learn their connection to polynomial solvability by radicals.
Understand groups where the upper central series reaches the whole group. Learn properties and examples of nilpotent groups.
Study groups with no relations except those forced by group axioms. Learn universal properties and applications to group presentations.
Explore connections to other areas of mathematics and advanced algebraic structures.
Generalize vector spaces to modules over arbitrary rings. Learn how this unifies many algebraic structures and provides powerful tools.
Understand how vector spaces are modules over fields. See how linear algebra theorems generalize to module theory over different rings.
Introduction to categories, functors, and natural transformations. See how algebra fits into the broader categorical framework of mathematics.
Study how abstract groups act as matrix groups. Learn characters, irreducible representations, and connections to other areas of mathematics.
Apply ring and field theory to number theory problems. Study rings of integers in number fields and unique factorization questions.
Focus on commutative rings and their ideals. Learn tools for algebraic geometry and deepen understanding of polynomial rings.
See how group theory, field theory, and number theory provide the mathematical foundation for modern cryptographic systems.
Explore how algebraic structures appear in geometry and topology. Understand symmetry groups and their role in classification problems.