172 
ARITHMETIC OF AN ALGEBRA [chap, x 
is a right divisor of every element of the first column, then 
d is equivalent to a matrix having the same first element and 
whose further elements in the first row and first column are 
all zero. 
For, if d l i = d 11 q{, we apply the transformation (ii) 
which adds to the elements of the ith column the products 
of those of the first column by k= — qi and find that the 
new ith element of the first row is zero. Similarly, if 
dii = Qid 11} we apply (i) with k= —Qi and find that the 
new ith. element of the first column is zero. 
Lemma 2. If the first element d xl of a matrix d is not 
zero and either is not a left divisor of every element of the 
first row or else is not a right divisor of every element of 
the first column, then d is equivalent to a matrix for 
which the first element is not zero and has a norm numer 
ically less than the norm of d xx . 
For, if du does not have d XI as a left divisor, property 
P shows that we can find elements q and r of S such 
that d l i = d xl q+r, where r^o and N(r) is numerically 
<N(d xl ). By (ii) we may add to the elements of the 
i\h column the products of those of the first column 
by — q and obtain an equivalent matrix having r as the 
ith element of the first row. By (iii) we obtain an equiva 
lent matrix having r as its first element. 
Similarly, if does not have d XI as a right divisor, 
we may write di 1 = Qd II -\-p, where p^o, and N{p) is 
numerically <N{d IZ ). We then use (i) with k= — Q. 
Bearing in mind that the norm of any element of S 
is an integer by property R which is zero only when the 
element is zero, we see that a finite number of applica 
tions of Lemma 2 leads to an equivalent matrix satisfying 
the hypothesis of Lemma 1. Hence any matrix d^ o is