Math 560
Spring, 2006
hw9

  1. Let p be a prime. Here are several facts about the field Zp. They are in fact theorems, but we don't have the time to prove them all. Definitely try to prove the ones I've marked with a (*). If you think you can, try to prove the others too. For each of the ones that you can't prove (with or without a *), show by example (or examples) that you understand it. (Z7 and Z11 are good places to look for examples.)

    1. (*) If a and b have square roots then so does a*b.

    2. (*) If a has a square root and b does not then neither does a*b.

    3. Exactly half the nonzero elements in Zp have square roots.

    4. If neither a nor b has a square root then a*b does have one.

  2. Suppose F is a field and r is an element of F that happens not to have a square root in F. For example, r might be -1 in the real numbers or -1=6 in Z7 or 2 in Z5. Let w be a symbol that is not the name of any element of F and consider the set G of all expressions of the form a+bw where a and b are in F. Define addition in the obvious way 
    	(a+bw) + (c+dw) = (a+c) + (b+d)w
    and multiplication using the distributive law and the assumption w2=r: 
    	(a+bw)(c+dw) = (ac+rbd) + (ad+bc)w

    Show that G is a field. To do that, first check all the ring axioms. Then show that when a+bw != 0+0w it has a multiplicative inverse. Be sure to point out where you use the fact that r does not have a square root in F.

  3. Why can't you use r=-1 in Z5 in the previous problem.

  4. Let r=-1 in F=Z3 in problem 2. Write out the addition and multiplication tables for the field G, which will have nine elements.

    Explain how you know that this ring with 9 elements is really not just Z9 (which also has 9 elements) in disguise, with the elements given different names.

  5. Let G be the field G constructed in problem 2 from F=Zprime and some r with no square root in F.

    1. Show that every element of F has a square root in G.

    2. Show that there are some elements of G that don't have square roots in G.

    As usual - if you can't prove something, show examples that show that you understand the statement clearly.