Galois theory is a beautiful area of mathematics which describes the relationship between field extensions, symmetry groups, and polynomials.  The most familiar example is complex conjugation.  Complex conjugation is an automorphism of the field of complex numbers, exchanging i and -i, which leaves the subfield of real numbers fixed.   This corresponds to exchanging the two roots of the polynomial x2 + 1.  Galois theory is about how permuting the roots of a polynomial corresponds to field automorphisms preserving a subfield.  Its most famous applications are to ruler and compass constructions (what figures are constructible?)  and the insolvability of a general quintic (fifth degree) equation.  It is named after Evariste Galois. a ninteenth century French mathematician. During the first part of the course, we will discuss field extensions, roots of polynomials, symmetric functions, the main theorem of Galois theory, and the applications mentioned above.  After that, we will take a detour into topology, and discuss how Galois theory is related to surfaces.