Math 102: Modern Algebra II (Section 1), Spring 2008

Math 102 Syllabus for Honors examiner

This is a brief description of the content of the course which will—if students are examined for honors—be the material transmitted to the honors examiner.

This and much more information on the course is available at the course website:
http://www.swarthmore.edu/NatSci/thunter1/Classes/102S08/index.html

Basic topic	Time Spent	Subtopics	Sections from Artin covered
Symmetry and Group actions	Two weeks	Motions of the plane and of space. Discrete subgroups of motions. Group actions on sets. Counting sizes of orbits.	Chapter 5—all.
More Group Theory	One week	The class equation. Simplicity of An. The Sylow theorems and some basic applications. Free groups and presentations of groups.	Chapter 6—all but section 9.
Bilinear forms	Two weeks	Sylvester's Law. Orthogonality. The spectral theorem. Conics and Quadrics. Normal operators. Skew-symmetric forms.	Chapter 7—all.
Group representations	Two weeks	G-invariant forms and Unitary representations. Irreducible representations and Mashke's theorem. Characters and the orthogonality relations. Permutation representations. One-dimensional representations. Schur's lemma.	Chapter 9—all but sections 3 and 10.
Abelian groups and modules over commutative rings	Two weeks	Free modules. Matrices and bases. Diagonalization of integer matrices. Finitely presented modules over the integers, and over PID's. Rational canonical and Jordan canonical form.	Chapter 12—all.
Fields	Two weeks	Algebraic and transcendental elements. Degree of an extension. Constructions with ruler and compass. Adjunction of roots. Finite fields. Function fields. Transcendental extensions and algebraic closure.	Chapter 13—all but section 7.
Galois Theory in characteristic zero.	Three weeks	The main theorem. Cubic and Quartic extensions of the rationals. Symmetric functions. Primitive elements. Kummer and cyclotomic extensions. Quintic extensions.	Chapter 14—all.