1. Prove Euclid I.6 : if two angles in a tiangle are equal to each other then the triangle is isosceles.
2. Suppose that the triangles ABC and DEF have |AB| = |DE|, |AC| = |DF| and the angle ABC equal the angle DEF. Does this imply that the triangles are congruent? (If yes prove it, if no give a counter-example.)
3. Solve the problems 1.10, 1.18, 2.14 from the book.
1.15, 1.16, 1.17, 2.20a, 2.26b
1. Prove that a group homomorphism f from a group G to a group H maps the identity element of G to the identity element of H and maps inverse elements to inverse elements (compare to Exercise 3.2).
2. Solve the problems 3.7, 3.15ab, 3.18.
4.5bc, 5.5, 5.6, 5.9
5.22, 5.25, 6.22, 6.24b, 6.27
1. On the elliptic curve y^2=(x-1)(x-2)(x-3), find all points P which satisfy the condition P+P=0 with respect to the group law + on the curve. Show that these points form a subgroup isomorphic to the Klein group V_4.
2. Use the algorithm from the proof of the Newton-Waring theorem to express r^4+s^4+t^4+u^4 as a polynomial in the elementary symmetric functions of the four variables r,s,t,u.
3. Solve the problems 7.5abc from the book.
7.8, 7.9, 7.10c, 7.12, 7.13
8.1, 8.6ab, 8.7, 8.10.
1. Show that the rotation group of the dodecahedron is isomorphic to the group A_5 of all even permutations of 5 symbols (it is called the alternating subgroup in S_5). Hint: the rotations permute 5 cubes formed by the diagonals in the pentagonal faces of the dodecahedron (each cube having one edge in each of the 12 faces of the dodecahedron).
2. Find the greatest common divisor of 50339 and 128243.
3. Referring to the picture on p. 105 of the book, find the ratio of the areas of the star ACEBD and the small pentagon FGHIJ.
4. Solve 10.13, 10.14.
11.7d, 11.16, 11.19, 11.20ab
12.6, 12.10, 12.11, 12.17b, 12.24
1. Prove that the group of units of any finite field is cyclic.
2. 13.13, 12.30abe