Vol. 4 No 9 - Pi Mu Epsilon

From (A) one RetS a = x.From (B) one gets eithera=xandb=yor a=yandb=x,If (A) and (B) then a = x and b = y.AN INTERESTING MAPPING OF TWO FIELDSIf (A) and (B) then a = x and b = y.Jerome M. KatzBrooklyn CollegeThen one has (a,b) = (x,y).In this article, I will prove an interesting theorem concerningCase - 2. {a\ = (,xy)a mapping of two fields, namely: If T is a one-to-one mapping of aand fa,b, = . -field F onto a field F' such that Tra(b-l)] = ~(a)[~(b) - ~(l)], thenFrom (C) a=x8= y and from (Dl a = b = x, which yield a = xT is an isomorphism.and b = y. Thus (a,b) = (x,y). This completes the proof.In the case where n = 3 (3-tuple or ordered triple), severaldefinitions will yield a result analogous to Theorem 1. For example,Definition 2. (a,b,c) = {{(a,c)l, {(a,b), (b,c)}}. (11)The reader can verify this definition by following the example ofTheorem 1. Attempts at other definitions will show that some apparentlyobvious ones are not sufficient.sufficient .(a,b,c) = { a} a b} {a,b c]}. Note that (1,1,2) #{1},{1,11,{1,1,2~} ; l{;},[l,2],[l,2,1~ since both sides,{1,2}} . This shows that the definition is notOne can extend this sort of definition to an n-tuple for any positiveinteger n. The following is a generalization of (I) and (11).Definition 3. (al,a2 ,..., a ) = {{(al,a2 ,..., a^,an)}, {(al,a2 ,...an-l), (a2,a3, ....a )I}. (111)It is not difficult to prove that (al,a2 ,..., a ) = (xl,x2 ,..., x ) if andonly if ({(al,a2, ..., a n-2dn )I, {(al,a2 ,..., a^), (a2+, ..., a)}} =In order to prove that T is an isomorphism, it is sufficient toshow that T preserves addition and multiplication. To do this, I willcharacterize addition and multiplication in terms of the operationa(b-1) which will be denoted a*b.Theorem 1: ab = (a*0) * r(l*Ol*bIProof:(a*O) * [(l*0)*b] = [a(O-111 * rl(0-l)*bl= (-a)*(-l*b)= (-a)*(-b+l)= (-a)(-b+l-1)= (-a)(-b) =ab{{(x1,x2,. . .xn_2,xn)1, {(x1,x2,. .X^), (x 2.x3....xn)}}.Note that this is a recursive definition of an ordered n-tuple, i.e.,we define an n-tuple in terms of sets whose elements are (n-1)-tuples.IWu i ng?BE SURE TO LET THE JOURNAL KNOW:Send your name, old address with zip code and newaddress with zip code to:If b=0, we obviously have a+b=a+O=a.We are given that T(a*b) = T [a(b-l)] = T(a) rT(b) - T(l)1 ;thus in order to prove that T(a*b) = T(a) * T(b), it is sufficient toprove that T(l) = 1' where 1' is defined to be the unity for F'.I will first prove that T(0) = 0' where 0' is the zero element ofF'.Theorem 3: If T is a onr-to-one mapping of a field F onto a field F'such that T [a(b-I)] = T(.,) [T(b) - T(l)], then T(0) = 0'.Proof:0 = O(a-1) for all a in F- T(0) = T [OCa-DlT(0) = T(0) CT(a) - T(l)1 (1)Assume T(0) # 0'. Then we can cancel T(0) from both sides of (1).Therefore, 1' = T(a) - T(l), and T(a) = T(l) + 1' for all a in F.Therefore, T maps every element of F into one element of F', acontradiction since T is one-to-one.Therefore T(0) = 0'.Pi Mu Epsilon Journal1000 Asp Ave. Rm. 215The University of OklahomaNorman. Oklahoma 73069