Galois Theory

The birth of Galois theory was originally motivated by the following question, which is known as the Abel-Ruffini theorem.

"Why is there no formula for the roots of a fifth (or higher) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc)?"

Galois theory not only provides a beautiful answer to this question, it also explains in detail why it is possible to solve equations of degree four or lower in the above manner, and why their solutions take the form that they do.

Galois theory also gives a clear insight into questions concerning problems in compass and straightedge construction. It gives an elegant characterisation of the ratios of lengths that can be constructed with this method. Using this, it becomes relatively easy to answer such classical problems of geometry as

"Which regular polygons are constructible polygons?"

"Why is it not possible to trisect every angle?"

-- wikipedia

 

The course

On Friday 23rd May 2008 Dr. Vassili Corbas (Reading University) will be giving a one day course at UCL on this fascinating subject. Dr. Corbas has many years experience of lecturing Galois theory and has a great interest in the subject. The course itself will focus on classical Galois theory with a view to understanding why it is that the general polynomial of degree greater than 4 is nonsoluable. It is being organised by Dr. Andrei Yafaev and Brian Tyler and is being funded by the Roberts Skills Training Fund.

The department is providing a free lunch in the college refectory for everyone who attends the course. Afterwards we might go to the union bar to continue on a more informal basis.

Course outline

Despite the time constraint, Dr. Corbas will be attempting to go over an entire lecture course in the space of three hours. In his own words:

"I will sketch proofs of points which are either critical or represent typical ways of thinking in the subject. But, having sketched the background, I would like to give complete proofs of the two punch line theorems: the fundamental theorem of solvability by radicals and that the general polynomial has the symmetric group as its Galois group."

If you want to do some reading prior to the course, then there is a free book by J.S Milne which covers the subject available here. The plan for the course is a follows:

First Hour

• Field Theory: (background reading 1, 2)
  • The existence of the splitting field.
  • The multiplicative property of dimensions.
  • The extension of isomorphisms to splitting fields.
  • Artin’s Lemma. Galois extensions: three equivalent conditions.
  • The Galois Correspondence and the Fundamental Theorem.
  • The example of X³− 2 over Q.
• Group Theory:
  • Solvable groups.
  • Simple criteria of solvability.
  • Examples.
  • Equations:
  • Solvability by Radicals.
  • Examples.

Second hour

• Cyclotomic extensions.
• Radical extensions and root towers.
• Cyclic and Abelian extensions.
• Lagrange’s Lemma.
• Normal closures.
• The Galois Theorem on solvability by radicals.

Third hour

• The general polynomial.
• The fundamental theorem of symmetric functions.
• The Galois group of the general polynomial is S_n.
• Brauer’s examples of specific non solvable rational polynomials.

 

Attending the course

The course is open to anyone who is interested in attending; there are no specific prerequisites for the course, although some undergraduate algebra would probably be helpful. The course is aimed primarily at a postgraduate mathematician audience, but should be accessible to third year undergraduates and theoretical physicists. The course will start at 12.00 and finish at 16.00 with a break for lunch at 13.00 and will take place in the mathematics department at UCL in room 807. See this map to find out where to go, the maths department is in square D1 and is located directly above the student union. Please email Dr. Andrei Yafaev or Brian Tyler if you are interested in attending - please note that there is no fee for this course.