Mathematics 582   Problem Set 4   Due Monday, October 28

1.  Find all of the irreducible monic polynomials of degree  2  in   Z3[l].   (There are three of them.)

2.   Let E  =  Z3[l]/(l2  + 1).  Let  r  be the element  l  + (l2  + 1)  in  E, so that  r  is a root of  l2  + 1.  (So,  E  is an extension field of  Z3,  and because the degree of  l2  + 1  is two, {1, r}  is a basis for  E  over  Z3.)  Write down all of the elements of  E  as linear combinations of  1  and  r.  How many elements are in  E?

3. Let  E*  be the multiplicative group of nonzero elements in  E.  Then  E*  is an Abelian group of order  8 (i.e., with  8  elements).  We also know that there are some Abelian groups of order  8  which are not cyclic.  An example is the additive group of the FIELD with  8 elements;  this is the field that we've been talking about in class.  Because the field with  8  elements has characteristic  2, the sum of any element with itself is  0, and no element has order  8  under addition.

(a)  Show that  r2 = 2  (this is the key to being able to multiply in  E.)

(b)  Show that  {1, r, r2, r3}  are distinct and that  r4 = 1.  This shows that  r  has order  4, and that if  E* were cyclic, then  r  could not be a generator of  E*.

4.  Show that the element  r + 1  in  E*  is an element of order  8.  (This proves that  E* is a cyclic group.)

5.  (a)  Because the dimension of E  over  Z3  is  2,  the set  { (r + 1)2,  r + 1, 1 }  is linearly dependent over  Z3.  Find a linear combination  a2(r + 1)2 + a1(r + 1)  +  a0  =  0,  in which  a2 = 1.

(b)  Which one of the irreducible polynomials that you found in problem  1  is the irreducible polynomial for  r + 1  in  Z3[l]?

(The underlying theorem, which we will not prove, is that if  E  is any finite field, the finite group E*  is cyclic.  A generator  s  of the group  E*  is called a primitive element of  E.  In our special case  s = r + 1.  In the general case, the characteristic of  E  is some prime number  p  and  E  is an extension field of  Zp.  The minimal polynomial in  Zp[l]  for  s  is called a primitive polynomial in  Zp[l].)