From the text:  Chapter 2, pages 43-47, #20, 24, 31, 32, 33, 34.

#24, page 44.  Let  G  be a free abelian group of rank  n  and let  H  be a subgroups.  Without loss of generality, we take  G = Z Å ... Å Z  (direct sum of  n  copies of  Z).  We will show, by induction on  n,  that  H  is a free abelian group of rank at most  n.  First prove it for  n = 1.  Then, assuming that the result holds for  n - 1,  let  p: G ® Z  denote the obvious projection of  G  onto the first factor (so that an n-tuple of integers gets sent to its first component).  Let  K  denote the kernel of  p.
    (a)    Show that  H Ç K  is a free abelian group of rank at most  n - 1.
    (b)    The image of  p(H) Í Z  is either  {0}  or infinite cyclic.  If it is  {0},  then  H = H Ç K;  otherwise, fix  h Î H  such that  p(h)  generates p(H)   and show that  H  is the direct sum of its subgroups  Zh  and  H ÇK.

#31, page 47.  Show that  (Ö3 + Ö7)/2  is an algebraic integer, hence the discriminant condition is actually necessary in Corollary 1, Theorem 12.

#32, page 47.  Find two fields of degree 3 over  Q,  whose composition has degree 6.

#33, page 47.  Let  w  be a primitive m-th root of unity for some  m ³ 3.  We know that  N_Q^Q[w](w) = ±1,  since  w  is a unit; show that in fact  N_Q^Q[w](w) = 1.

#34, page 47.  Let  w  be a primitive p^t-th root of unity for some prime  p  and positive integer  t.
    (a)    If  k  is a positive integer relatively prime to  p, show that  1 + w+ w² ...  + w^k-1  is a unit in  Z[w].  (HINT:  Its inverse is  (1 - w) / (1 - w^k).  Show that  w = w^hk  for some integer  h.)
    (b)    Show that  p = u(1 -w)ⁿ  for some unit  u Î Z[w],  where  n = f(p^t).
    (c)    Show that  N_Q^Q[w](1 - w) = p.
    (d)    Show that  disc(w) = p^m  where  m = p^t-1(pt - t - 1).