## Tutorial 1 : Algebraic Extensions and Ruler and Compass Constructions

1. Let be an extension of degree
(a) For any prove that the map defined by for all is a linear transformation of the -vector space Show that is isomorphic to a subfield of the ring of matrices with entries in
(b) Prove that is a root of the characteristic polynomial of Use this procedure to find monic polynomials satisfied by and

2. Prove that is not a sum of squares in the field where

3. Let be a field and be an indeterminate. Find the irreducible polynomial of over where

4. Find an algebraic extension of such that the polynomial has a root in

5. The construction of a regular -gon amounts to the construction of the real number Show that is a root of Hence conclude that a regular -gon is not constructible by ruler and compass.

6. Show that an angle of degrees, is constructible if and only if

7. Prove that it is impossible, in general, to quintsect an arbitrary angle by ruler an compass. Is it possible to divide the angle degrees into five equal parts by ruler and compass ?