FORMAL KEALIZAIULITY THEORY II 583 



cannot remo\'e from any Ek(p) a factor which was present in carUer 

 Ej{p)(j < A) but was not cancelled therefrom (treat each linear factor 



of ^z and of Ki as distinct, and each linear factor of ^,^ as distinct 



Ek(p) 



to see this easily). 



Property (e') is best proved by a reductio ad absurdum. We recall 

 that Ei(p) is the highest common factor of all elements of '^yXp)^(p)- 

 Suppose now that Ei{p) contained a factor <p in common with ^z(p). 

 Then every non-zero element of ^z(p)Z(p) contains the factor <p. Hence 

 no denominator in Z(p) cancels (p from ^z(?>). Hence no denominator 

 contains ^ as a factor, but this denies its presence in their least common 

 multiple, ^z(p). 



The uniqueness of W{p) follows at once from the uniqueness lemma, 

 0.11. Multiply (2) by ^z(p). Then 



^z(p)W\p) = A\p)^z(p)Z(p)B\p) (3) 



lias diagonal elements of the form 



'^z(p)ekip) 

 ^kip) ' 



1 < k < R. (4) 



But by (3) and (d'), these are the result of polynomial operations on 

 the polynomial matrix ^z(p)Z(p). Hence the elements (4) are poly- 

 nomials, and each has unit leading coefficient. ■^z(p)TF°(p) then clearly 

 satisfies (a), (b), (c), and (d) of 5.1. Therefore by 5.11, '^z(p)W\p) = 

 Eip) = ^z{p)W(p). Therefore W\p) = W(p). Q.E.D. 



5.14 Corollary: W{p) is its own normal form. 



5.15 Corollary: Let (p{p) be a rational function and 



Z^{p) = <p(p)Z(p). 



\.vt W(p) be the normal form of Z(p) and Wi{p) the normal form of 

 Zi(p). Then, when written in normalized lowest terms, 



TFi(7>) = <p(p)W{p). 



Proof: Supposing that (2) above holds for W and Z, we have 



^(p)W(p) = A(p)Z,(p)B(p). 



Call the left side of this equation Wl{p). We must identify this ^\•illl 

 ITiCp). We have just showed that it satisfies (d') of 5.13. It clearly 

 satisfies (a'), (b') and (c'), with Zi written for Z. Hence 5.13 implies the 

 desired equality. 



