Squaring the Circle and Irreducible Polynomials

Module code: MA4103

This module studies the irreducibility of polynomials and how irreducible polynomials are used to construct fields. This theory is then applied in an elegant way to the three classical Greek problems of ‘Squaring the circle’, ‘Duplicating the cube’, and ‘Trisecting an angle’. These problems were being studied, as the name suggests, by the Ancient Greeks in 4th century BCE, but remained unsolved until the 19th century, when algebraic methods were applied.

The module introduces some basic ring theory and investigates the connections between factor rings and fields. You will then move on to discuss properties of polynomials over fields, especially the rationales, with the focus on irreducibility. This enables us to determine when a factor ring is a field, and hence to construct many new fields (i.e. other than Q, R, C and Zp for p prime). We then move on to field theory and extension fields, and show how all finite fields can be constructed. Finally, we bring all the theory together with the study of ruler and compass constructions, and give a simple and neat solution to each of the three classical Greek problems mentioned above.