Galois Theory

Course Objective

• The student knows the following concepts, and can solve problems about and with them in explicit situations:
• basic field theory (various properties of field extensions, including finiteness, algebraicity, separability and normality);
• splitting fields, finite Galois extensions and the Galois correspondence;
• cyclotomic extensions of the rationals;
• finite fields, as well as their finite extensions together with their Galois groups.
• The student knows the criteria for constructibility by straightedge and compass, and solvability by radicals, and can apply those in explicit situations.

Course Content

The Babylonians could solve quadratic equations, but it took until the 16th century before (complicated) formulae were discovered for solving cubic and quartic equations. These use iterated expressions of (varying) n-th roots ('radicals'). The quest for similar formulae for higher degree equations culminated in a negative answer by Ruffini and Abel in the early 19th century. Galois soon after refined the result, providing a precise criterion in terms of the symmetries of the zeroes of the equation that determines if those zeroes can be expressed using radicals.

After a more general study of field extensions and their properties, we discuss the fundamental theorem of Galois theory, one of the most beautiful results in algebra. We also discuss its consequences for the (in)solvability of the quintic by radicals, and for the (in)constructibility of regular n-gons by means of straightedge and compass.

We treat the following topics.

• Properties of field extensions (including finite, algebraic, simple, separable, inseparable, normal or finite Galois extensions); examples.
• Compositum of fields; algebraic closure (including a different proof of the fundamental theorem of algebra, which asserts that the complex numbers are algebraically closed).
• Cyclotomic fields.
• The fundamental theorem of Galois theory, which gives a 1-1 correspondence between the subgroups of the Galois group of a finite Galois extension, and the intermediate fields.
• Finite fields, as well as their finite extensions together with their Galois groups.
• Constructibility using straightedge and compass; (in)solvability of polynomials using radicals.

Additional Information Teaching Methods

Lectures (14x2 hours) and tutorials (14x2 hours)