In this course we solve some of the classical geometric problems such as "squaring the circle", "which angles are trisectable", and "which regular p-gons are constructible" going back to the ancient greeks but solved only about 2000 years later!
To this end we systematically analyse fields and field extensions (field theory).
This algebraic theory is very powerful and allows us, to deduce many more interesting results, including a complete classification of all finite fields, and a (much more precise) analogue of the prime number theorem in polynomial rings over finite fields.