Algebra Qualifying Syllabus
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.
Matrices
Free modules
Definition: An \(R\)-module \(M\) is a free \(R\)-module if \(M\) has a basis. A subset \(A\) of an \(R\)-module \(M\) is a basis of \(M\) if \(A\) is linearly independent and generates \(M\).
Examples:
- \(R=R\{1_R\}\) is a free \(R\)-module
- \(R^2=R\oplus R\) has basis \(\{(1_R,0_R),(0_R,1_R)\}\)
- \(R[x]\) is a free \(R\)-module with (infinite) basis \(\{1,x,x^2,...,x^i,...\}\)
- \(R[x,y]\) has basis \(\{x^n,y^m\ |\ n,m\geq 0 \}\) as free \(R\)-module
- \(R[x,y]\) has basis \(\{1,y,y^2, ...,y^i,...\}\) as free \(R[x]\)-module
All free \(R\)-modules \(M \) have some basis \(B=\{b_1,...,b_n\} \) so have rank \(n \), and can be written as \(R^n\cong M \). Additionally every element \(m\in M \) can be uniquely written \( m=\sum_{i=1}^n r_ib_i\).
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.
