Homework Assignment #2
Due in class on Friday, 1/30/04

From the text:  Chapter 2, page 39, #1, 2, 3, 4, 5, 6, and the exercise on page 16.

#1, page 39.
    (a)    Show that every number field of degree 2 over  Q  is one of the quadratic fields  Q[Öm], m Î Z.
    (b)    Show that the fields  Q[Öm], m  square-free, are pairwise distinct.  (HINT:  Consider the equation  Öm = a + bÖn; use this to show that they are in fact pairwise non-isomorphic.)

#2, page 39.  Let  I  be the ideal generated by  2  and  1 + Ö-3  in the ring  Z[Ö-3] = {a + bÖ-3 : a,b Î Z}.  Show that  I ¹ (2)  but  I 2 = 2I.  Conclude that ideals in  Z[Ö-3]  do not factor uniquely into products of prime ideals.  Show, moreover, that  I  is the unique prime ideal containing  (2),  and conclude that  (2)  is not a product of prime ideals.

#3, page 39.  Complete the proof that, if  m  is a square-free integer, then the set of algebraic integers in the quadratic field  Q[Öm]  is

{a + bÖm : a,b Î Z}  if  m º 2 or 3 (mod 4),
{(a + bÖm)/2 : a,b Î Z, a º b (mod 2)}  if  m º 1 (mod 4).

#4, page 39.  Suppose  a0,...,an-1  are algebraic integers and  a  is a complex number satisfying

an + an-1 an-1 + ... + a1a + a0 = 0

Show that the ring  Z[a0,...,an-1,a]  has a finitely generated additive group.  (HINT:  Consider the products  a0m0...an-1mn-1 am  and show that only finitely many values of the exponent are needed.)  Conclude that a  is an algebraic integer.

#5, page 39.  Show that, if  f  is any polynomial over  Zp = Z / pZ  (where  p  is a prime), then  f(xp) = (f(x))p.  (Suggestion:  Use induction on the number of terms.)

#6, page 39.  Show that, if  f  and  g  are polynomials over a field  K  and  f 2 | g  in  K[x], then  f | g'.  (HINT:  Write  g = f 2h  and differentiate.)

Exercise, page 16.  Using the determinant procedure of Theorem 2, find monic polynomials (of degree six) satisfied by the algebraic integers  Ö2 + 3Ö3  and  Ö2 × 3Ö3  (the sum and the product of the square root of 2 and the cube root of 3).