Let alpha Beta element of Z3 be zeros of x2 1 and x2 x 2

Let alpha, Beta element of Z_3 be zeros of x^2 + 1 and x^2 + x + 2 in Z_3[x], respectively. Prove that the two fields^Z_3(alpha) and^Z_3(beta) are isomorphic. Are they equal and why?

Solution

given that

The fact that X and Y are linearly isomorphic to each other and to R n are basic facts from linear algebra which we assume. From the latter, it is sufficient to prove the topological equivalence result for the case X D Y D R n . We will show that the norm k kX is equivalent to a specific norm k k1 which is easier to work with, and the general result will follow by a transitivity argument. Let fekg n kD1 be the standard basis of R n . We define for every x 2 R n , kxk1 D Xn kD1 xkek 1 D max i fjxi jg: This expression is easily seen to satisfy all of the norm axioms. We can also see that this norm is equivalent to the usual Euclidean norm: kxk1 Xn kD1 jxkj 2 !1=2 Xn kD1 max i jxi j 2 !1=2 n 1=2kxk1: In particular, any set that is compact with respect to the usual Euclidean norm is compact with respect to k k1. 2 1 Introduction For general k kX we have the basic estimate kxkX D Xn kD1 xkek X Xn kD1 jxkjkekkX Xn kD1 kxk1 max i fkeikX g D n max i fkeikX g kxk1: (1.2) This leads us to define X D n max i fkeikX g > 0: It follows from this estimate that the function W .R n ; k k1/ ! R given by .x/ D kxkX is continuous. Indeed, if x; y 2 R n then j.x/ .y/j D jkxkX kykX j kx ykX X kx yk1: The boundary of unit ball with respect to k k1, @B1.0; 1/, is closed and bounded and hence, by the Heine-Borel theorem, compact. The continuous function thus attains its minimum at some point x0 2 @B1.0; 1/. As 0 62 @B1.0; 1/ we cannot have .x0/ D kx0kX D 0, so that X D .x0/ > 0: Then for any x 2 R n we have x kxk1 X X H) kxkX X kxk1 (1.3) Combining (1.2) and (1.3) gives that for any x 2 R n , X kxk1 kxkX X kxk1: If k kY is any other norm a calculation shows that X Y kxkY kxkX X Y kxkY and hence the norms k kX and k kY are equivalent.