From the text:  Chapter 4, pages 114-119, #1, 6, 7, 8, 9, 13.

#6, page 116.  Let  m  and  n  be distinct square-free integers, neither equal to 1.  The biquadratic field  K = Q[Öm,Ön]  is a normal extension of  Q  with Galois group isomorphic to the Klein 4-group (direct sum of two cyclic groups of order 2).  Thus, there are three quadratic subfields (what is the third?).  Let  p  be a prime in  Z.
    (a)    Suppose that  p  is ramified in each of the quadratic subfields.  What happens in  K?  Find an example.
    (b)    Suppose that  p  splits completely in each of the quadratic subfields.  What happens in  K?  Find an example.
    (c)    Suppose that  p  is inert in each of the quadratic subfields.  What happens in  K?  Can this ever occur?
    (d)    Find an example in which  p  splits into  PQ  in  K;  P²Q²;  P².

#7, page 116.  Find a prime  p  and quadratic extensions  K  and  L  of  Q  illustrating each of the following.
    (a)    p  can be totally ramified in each of  K  and  L  without being totally ramified in  KL.
    (b)    K  and  L  can each contain unique primes lying over  p  while  KL  does not.
    (c)    p  can be inert in  K  and  L  without being inert in  KL.
    (d)    The residue field extensions of  Z_p  can be trivial for  K  and  L  without being trivial for  KL.

#8, page 116.  Let  e,  f,  and  g  be given positive integers.
    (a)    Show that there exist primes  p  and  q  such that  p  splits into  g  distinct primes in the  q-th cyclotomic field  K.
    (b)    Show that  p  and  q  in part (a) can be taken so that  K  contains a subfield of degree  fg  over  Q.  How does  p  split in this subfield?
    (c)    Show that the further condition  p º 1  (mod e)  can be satisfied.
    (d)    Show that,  with  p  and  q  as above, the pq-th cyclotomic field contains a subfield in which  p  splits into  g  primes, each with ramification index  e  and inertial degree  f.
    (e)    Find an example of  p  and  q  for  e = 2,  f = 3,  g = 5.

#9, page 117.  Let  L  be a normal extension of  K,  P  a prime of  K,  Q  and  Q'  primes of  L  lying over  P.  We know that  Q' = s(Q)  for some  s Î G.  Let  D  and  E  be the decomposition and inertia groups for  Q  over  P,  and  D'  and  E'  the decomposition and inertia groups for  Q'  over  P.
    (a)    Prove that  D' = sDs^-1,  E' = sEs^-1.
    (b)    Assuming further that  P  is unramified in  L,  the Frobenius automorphisms  f = f(Q|P)  and  f' = f(Q' |P)  are defined.  Prove that  f' = sfs^-1.

#13, page 119.  Let  m Î Z,  and assume that  m  is not a square.  Then  K = Q[⁴Öm]  has degree 4 over  Q,  and  L = Q[⁴Öm,i]  is its normal closure over  Q.  Setting  a = ⁴Öm  and denoting the roots  a,  ia,  -a,  -ia  of  x⁴ - m  by the numbers 1, 2, 3, 4, respectively, we can represent the Galois group  G  of  L  over  Q  as permutations of 1, 2, 3, 4.
    (a)    Show that  G = {1, t, s, ts, s², ts², s³, ts³},  where  t  is the permutation  (2,4)  (meaning  2 ® 4 ® 2,  with 1 and 3 fixed), and  s = (1,2,3,4)  (meaning  1 ® 2 ® 3 ® 4 ® 1).
    (b)    Suppose that  p  is an odd prime not dividing  m.  Prove that  p  is unramified in  L.
    (c)    Let  Q  be a prime of  L  lying over  p  (with  p  as in (b)), and suppose that  f(Q|p) = t.  Use Theorem 33 to show that  p  splits into three primes in  K.
    (d)    Determine how  p  splits in  K  for each of the possibilities for  f(Q|p).