Math 3510 or an undergraduate course in abstract algebra.
Students are expected
to be familiar with groups and be comfortable with algebraic formalism
and abstraction.

Course Description

A set of numbers with addition, subtraction, multiplication, and division
is called a field. Familiar examples are the rational numbers Q,
the real numbers R, and the complex number C. A less familiar
example is, say, the Gaussian rational numbers Q[i]={a+bi : a,b are in Q}. A field, such as this last example, which is
also a finite-dimensional Q-vector space is called a
number field. The automorphisms of a number field form a group,
and there is a beautiful connection between the structure of this group
and the algebraic and arithmetic properties of the number field.
This is what is known as Galois Theory, after its inventor,
Evariste Galois. This course will be a concrete hands-and exploration
of this so-called Galois correspondence, and we will make extensive use
of computer explorations. This course serves as a bridge between an
undergraduate first-course in abstract algebra and a graduate-level
modern algebra course.

Textbook

John Swallow,
Exploratory Galois Theory, Cambridge University Press, 2004