Algebra Qualifying Syllabus - Revision history

Kfagerstrom: /* Modules: */

2023-04-26T16:26:05Z

Modules:

Kfagerstrom at 16:17, 26 April 2023

2023-04-26T16:17:22Z

Kfagerstrom at 18:01, 10 March 2023

2023-03-10T18:01:17Z

Kfagerstrom at 01:25, 8 March 2023

2023-03-08T01:25:30Z

Kfagerstrom at 01:05, 8 March 2023

2023-03-08T01:05:07Z

Kfagerstrom: Created page with "'''Group Theory:''' Groups, subgroups, homomorphisms, cosets, quotients, isomorphism theorems, direct and semidirect products, solvable groups, structure of cyclic, symmetric..."

2023-03-08T01:03:51Z

Created page with "'''Group Theory:''' Groups, subgroups, homomorphisms, cosets, quotients, isomorphism theorems, direct and semidirect products, solvable groups, structure of cyclic, symmetric..."

New page

'''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.

← Older revision		Revision as of 18:01, 10 March 2023
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	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.
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		+	===random pages to be categorized===
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		+	[[Cayley-Hamilton Theorem]]

@@ Line 17: / Line 17: @@
 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:==

@@ Line 1: / Line 1: @@
-'''Group Theory:''' Groups, subgroups, homomorphisms, cosets, quotients, isomorphism theorems, direct and semidirect products, solvable groups, structure of cyclic, symmetric and
+==Group Theory:==
 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
+==Group Actions:==
 automorphism groups, p-subgroups and the Sylow Theorems.
-'''Ring Theory:''' Definition and examples, homomorphisms, ideals, quotients, integral domains
+==Ring Theory:==
 and their fields of fractions, maximal and prime ideals.
 Factorization in Commutative Rings: Euclidean domains, Unique Factorization Domains,
@@ Line 11: / Line 14: @@
 Gaussian integers.
-'''[[Modules]]:''' Definition and examples: matrices, free modules, and bases over arbitrary commutative rings, Structure Theorem for finitely generated modules over PIDs, applications
+==[[Modules]]:==
 to linear operators: Jordon and rational canonical forms.
-'''Basic Linear Algebra:''' vector spaces, bases, dimension, bases for infinite dimensional spaces
+==Basic Linear Algebra:==
 (Zorn’s Lemma), linear transformations, eigenvectors, characteristic polynomial, diagonalization, Cayley-Hamilton Theorem.
-'''Field Theory:''' Definition and examples, algebraic and transcendental extensions, degree of
+==Field Theory:==
 a finite extension, multiplicativity of degrees, adjunction of roots, finite fields.
-'''Basic Galois Theory for Finite Separable Extensions:''' Definitions of Galois group and
+==Basic Galois Theory for Finite Separable Extensions:==
 Galois field extensions, the main theorem of Galois theory, primitive elements, Kummer
 extensions, cyclotomic extensions, quintic polynomials.