Difference between revisions of "Algebra Qualifying Syllabus"
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| − | + | ==Group Theory:== | |
| + | Groups, subgroups, homomorphisms, cosets, quotients, isomorphism theorems, direct and semidirect products, solvable groups, structure of cyclic, symmetric and | ||
alternating groups; free groups, structure theorem for finite abelian groups. | alternating groups; free groups, structure theorem for finite abelian groups. | ||
| − | + | ==Group Actions:== | |
| + | Groups acting on sets, cosets, and themselves; orbits and stabilizers, permutation representations, Cayley’s Theorem, the class equation, inner automorphisms and | ||
automorphism groups, p-subgroups and the Sylow Theorems. | automorphism groups, p-subgroups and the Sylow Theorems. | ||
| − | + | ==Ring Theory:== | |
| + | Definition and examples, homomorphisms, ideals, quotients, integral domains | ||
and their fields of fractions, maximal and prime ideals. | and their fields of fractions, maximal and prime ideals. | ||
Factorization in Commutative Rings: Euclidean domains, Unique Factorization Domains, | Factorization in Commutative Rings: Euclidean domains, Unique Factorization Domains, | ||
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Gaussian integers. | Gaussian integers. | ||
| − | + | ==[[Modules]]:== | |
| + | Definition and examples: matrices, free modules, and bases over arbitrary commutative rings, Structure Theorem for finitely generated modules over PIDs, applications | ||
to linear operators: Jordon and rational canonical forms. | to linear operators: Jordon and rational canonical forms. | ||
| − | + | ==Basic Linear Algebra:== | |
| + | vector spaces, bases, dimension, bases for infinite dimensional spaces | ||
(Zorn’s Lemma), linear transformations, eigenvectors, characteristic polynomial, diagonalization, Cayley-Hamilton Theorem. | (Zorn’s Lemma), linear transformations, eigenvectors, characteristic polynomial, diagonalization, Cayley-Hamilton Theorem. | ||
| − | + | ==Field Theory:== | |
| + | Definition and examples, algebraic and transcendental extensions, degree of | ||
a finite extension, multiplicativity of degrees, adjunction of roots, finite fields. | a finite extension, multiplicativity of degrees, adjunction of roots, finite fields. | ||
| − | + | ==Basic Galois Theory for Finite Separable Extensions:== | |
| + | Definitions of Galois group and | ||
Galois field extensions, the main theorem of Galois theory, primitive elements, Kummer | Galois field extensions, the main theorem of Galois theory, primitive elements, Kummer | ||
extensions, cyclotomic extensions, quintic polynomials. | extensions, cyclotomic extensions, quintic polynomials. | ||
Revision as of 16:17, 26 April 2023
Contents
Group Theory:
Groups, subgroups, homomorphisms, cosets, quotients, isomorphism theorems, direct and semidirect products, solvable groups, structure of cyclic, symmetric and alternating groups; free groups, structure theorem for finite abelian groups.
Group Actions:
Groups acting on sets, cosets, and themselves; orbits and stabilizers, permutation representations, Cayley’s Theorem, the class equation, inner automorphisms and automorphism groups, p-subgroups and the Sylow Theorems.
Ring Theory:
Definition and examples, homomorphisms, ideals, quotients, integral domains and their fields of fractions, maximal and prime ideals. Factorization in Commutative Rings: Euclidean domains, Unique Factorization Domains, Principal Ideal Domains, Gauss’s Lemma, polynomial factorization, Eisenstein’s criterion, Gaussian integers.
Modules:
Definition and examples: matrices, free modules, and bases over arbitrary commutative rings, Structure Theorem for finitely generated modules over PIDs, applications to linear operators: Jordon and rational canonical forms.
Basic Linear Algebra:
vector spaces, bases, dimension, bases for infinite dimensional spaces (Zorn’s Lemma), linear transformations, eigenvectors, characteristic polynomial, diagonalization, Cayley-Hamilton Theorem.
Field Theory:
Definition and examples, algebraic and transcendental extensions, degree of a finite extension, multiplicativity of degrees, adjunction of roots, finite fields.
Basic Galois Theory for Finite Separable Extensions:
Definitions of Galois group and Galois field extensions, the main theorem of Galois theory, primitive elements, Kummer extensions, cyclotomic extensions, quintic polynomials.
