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, | ||
| Line 11: | Line 14: | ||
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. | ||
| + | ===Matrices=== | ||
| + | ===Free modules=== | ||
| + | Definition: An <math>R</math>-module <math>M</math> is a '''free''' <math>R</math>-module if <math>M</math> has a basis. A subset <math>A</math> of an <math>R</math>-module <math>M</math> is a basis of <math>M</math> if <math>A</math> is linearly independent and generates <math>M</math>. | ||
| − | + | ====Examples:==== | |
| + | # <math>R=R\{1_R\}</math> is a free <math>R</math>-module | ||
| + | # <math>R^2=R\oplus R</math> has basis <math>\{(1_R,0_R),(0_R,1_R)\}</math> | ||
| + | # <math>R[x]</math> is a free <math>R</math>-module with (infinite) basis <math>\{1,x,x^2,...,x^i,...\}</math> | ||
| + | # <math>R[x,y]</math> has basis <math>\{x^n,y^m\ |\ n,m\geq 0 \}</math> as free <math>R</math>-module | ||
| + | # <math>R[x,y]</math> has basis <math>\{1,y,y^2, ...,y^i,...\}</math> as free <math>R[x]</math>-module | ||
| + | |||
| + | |||
| + | All free <math>R</math>-modules <math>M </math> have some basis <math>B=\{b_1,...,b_n\} </math> so have rank <math>n </math>, and can be written as <math>R^n\cong M </math>. Additionally every element <math>m\in M </math> can be uniquely written <math> m=\sum_{i=1}^n r_ib_i</math>. | ||
| + | |||
| + | |||
| + | |||
| + | |||
| + | ===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. | (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. | ||
| + | |||
| + | |||
| + | ===random pages to be categorized=== | ||
| + | |||
| + | [[Cayley-Hamilton Theorem]] | ||
Latest revision as of 16:26, 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.
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.
