# Math 521 ``Graduate Algebra I''

## Fall 2011

**Professor:**
Eric Brussel
(Room W414; 7-5605;
Email *brussel* at *mathcs.emory.edu*
)
**Text:**
*Algebra* by Dummit & Foote

**Class:**
MSC E408, TTh 10-11:15

**Introduction:**
Abstract Algebra is the branch of pure mathematics
that tries to reduce mathematical questions to symbolic manipulation.
Here is an example. Consider a cube, an object most
people feel they understand. 6 congruent square faces,
12 congruent edges, 8 vertices.
How much symmetry does it have? This question seems to
make sense already, but it is hard to quantify until we define
``symmetry''.
In algebra, a *symmetry* of a geometric object
is a bijection of the object to itself that preserves the object's defining
properties.
That is an algebraic formulation of
a geometric concept.
``Preserving the object's defining properties'' in this case means
preserving the cube's shape (metric) and its orientation in space.
So, how many geometric symmetries does the cube have?
Our position in this class is that
we don't really understand the cube
if we cannot answer this question.
Our collective mind will *fester* around this problem,
until we have built up the machinery necessary to answer it.
We will see that not only can we count the symmetries,
we can describe the *structure* of this set, using the algebraic
notion of *group*. We will *name* the *symmetry
group* of the cube.

For the last few hundred years,
algebra has developed mostly around attempts to
solve specific math problems, mostly in number theory and geometry.
For example, algebra was used to solve the
ancient problem
of determining which polygons can be constructed using only a
straight edge and compass.
Gauss solved this problem in 1796, when he was five years old.
We'll try to solve some of these types of problems.

``Abstract'' means, ``disassociated from any specific instance''.
In the 20th century, certain basic algebraic objects were isolated
and abstracted to form the core of abstract algebra:
groups, rings, fields, and modules.
These abstract objects appear in practically every area of pure mathematics.
We'll start with groups, then start on rings.
We'll do modules and fields in Math 522.

**Prerequisites:**
Undergraduate Abstract Algebra

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Brussel's page