Section III, 1918 



[139] 



Trans. R.S.C. 



On the Embodiment in Actual Numbers of the Kummer Ideals in the 



Quadratic Realm 



By J. C. Glashan, LL.D., F.R.S.C. 

 (Read May Meeting, 1918). 



The postulates defining the integral domain in the general quad- 

 ratic realm are: — (1) The span of every quadratic integer is a rational 

 integer, (2) The norm of every quadratic integer is a rational integer. 

 The limitation imposed by the first postulate causes the failure in 

 the integral quadratic domain of the unique-factorization law. The 

 introduction of the Kummer ideals restores this law but necessitates 

 the modification of the first postulate to : The spans of the bases of 

 every ideal are rational integers. The adjunction of an auxiliary 

 radical to the fundamental radical permits of the determination of 

 actual numbers which restore the unique-factorization law and re- 

 place ideals in every other respect. The first postulate delimiting 

 these biquadratic integers becomes: the product of the span of every 

 biquadratic integer and its auxiliary radical is a rational integer. 



