This course will provide an introduction to algebraic structures and modern basic algebra. The main purpose of the course is to broaden the perspective and understanding of high school teachers by linking topics in high-school algebra with modern algebra. This kind of understanding and perspective is essential for teachers in planning high school syllabi and activities. In particular, it should allow them to focus more on the "big ideas" in algebra rather than concentrating mainly on developing algebraic techniques and skills. The high-school curriculum leaves students the impression that algebra is involved mainly in the solution of different types of equations. Many remain with this impression even after having learned linear algebra at the university level. It is certainly true that the desire to solve certain types of equations gave rise to modern algebra, however, modern algebra has a completely different viewpoint, and is concerned with algebraic structures and operations.
The two most important topics in high-school algebra are arguably the solution of polynomial equations and systems of linear equations. These connect directly to some of the most important and fundamental theorems in algebra. In particular, we shall follow the attempts of mathematicians to solve polynomial equations which resulted in the development of group and field theory by Galois and Abel, and review topics and the fundamental theorems in both areas. We shall take a historical perspective, using some popular mathematical texts (see the bibliography below) as a starting point, and "filling in" the mathematical content. We shall include many proofs whenever they are accessible to the audience and further their understanding. Our aim is not to prove the deep results mentioned but to develop enough tools to discuss and understand them. Solving systems of linear equations led to the development of linear algebra, and we shall review theorems in linear algebra and some applications in coding theory.
All the topics covered in the course will be constructed around examples that shall be investigated in detail, thus adding to the teachers' knowledge a number of interesting and basic examples and applications in algebra. We shall also present some of the major recent results and directions that current research in algebra is taking. It is important to realize that apart from being an interesting and fruitful area for mathematical research, algebra continues to have applications in many areas - often quite unexpectedly - and this will be illustrated with recent examples.
Only basic knowledge of linear algebra will be assumed. More advanced material will be defined and reviewed so that the course can also be considered an introduction to algebraic structures for those who have not previously studied them.
Topics to be covered:
- Review of algebraic structures: rings, fields, groups, vector spaces. Substructures, detailed examples
- Ring Theory: ideals, homomorphism, and quotients. commutative rings, examples, special types of rings. The Chinese Remainder Theorem and its application to RSA coding.
- Group theory: Introduction, subgroups, quotient groups. Sylow's theorems. Detailed examples.
- Field theory: Introduction. Extensions of fields.
- Cyclotomic fields, Galois groups, detailed examples.
- Finite fields: Detailed examples and properties.
- The general polynomial equation of degree n and solvability by radicals.
- - Constructability by ruler and compass.
- - Squaring the circle, doubling the cube.
- - Vector spaces: Vector spaces over fields of characteristic 2, and their uses in error-correcting codes.
- J. Bergen: A Concrete Approach to Abstract Algebra: From the Integers to the Insolvability of the Quintic, Elsevier, Academic Press, 2010
- J. Derbyshire: Unknown Quantity: A real and imaginary history of Algebra, Atlantic Books, London, 2008
- L. Gaal: Classical Galois Theory, Chelsea Publishing Co. NY, 1973
- I.M. Isaacs: Algebra: A Graduate Course,
- N. Jacobson: Basic Algebra I
- S. Lang: Algebra
- M. Livio: The Equation that couldn't be solved, Simon & Schuster Paperbacks, NY, 2006 בעברית : "שפת הסימטריה: המשוואה שלא נמצא לה פתרון",
- S. Singh: The Code Book, Anchor Books, Random House Inc., NY, 1999