333 SECTIONAL TRANSACTIONS —A*. 



DEPARTMENT OF MATHEMATICS (A*). 



Thursday, September 10. 



Dr. Olga Taussky. — Modern problems in algebraic number theory (io.o). 



A few decades ago Hilbert emphasised the connection between the 

 abelian extension fields of an algebraic number field and the division of the 

 ideals of the base field into classes of equivalent ideals. Since then class 

 field theory has become the main topic of algebraic number theory. Although 

 class field theory is restricted to abelian fields only, two of the most funda- 

 mental questions of the general theory can be reduced to it. The first is 

 the problem of enumerating all the extension fields of a field K in which the 

 ideals of K become principal ideals. Furtwangler's principal ideal theorem 

 asserts that the Hilbert class field is one of these extension fields. The 

 second question, which is still unsolved, is whether there exists an extension 

 field for every algebraic number field which contains only principal ideals. 

 This question can easily be shown to be equivalent to the so-called class field 

 tower problem, i.e. the problem whether the sequence K = K , K 1} . . ., 

 K n , . . ., where K% is the class field of Ki- X , ends after a finite number of 

 elements. Hilbert conjectured almost all the properties of class fields, but 

 to prove his statements was by no means an easy task. That is particularly 

 the case for the principal ideal theorem. This theorem is proved by means 

 of a theorem on abstract finite groups. All the proofs of it which have been 

 given use the methods of modern algebra. Using abstract group theory 

 it is possible in some cases to prove the finiteness of the class field tower by 

 a close investigation of the Hilbert class field and its subfields. 



Dr. J. Gillis. — Some notes on the modern theory of measure (n.o). 



Linearly measurable plane sets are defined and their main known pro- 

 perties described. This leads to the division of these sets into two cate- 

 gories — regular and irregular. The former have all the fundamental 

 properties of rectifiable curves while the latter are fundamentally different 

 from them. It is irregular sets that are discussed here. 



(i) It has been conjectured that such sets have zero projection (i.e. 

 projection of zero measure) on almost all directions. A description is given 

 of such parts of this conjecture as have actually been proved, including some 

 hitherto unpublished results, and a discussion of their possible extension 

 follows. 



(2) It was known that, at almost all points of an irregular set, the upper 

 density in every angle is positive. The problems which arise in the case of 

 the lower density are discussed in relation to the known facts. 



Dr. T. Estermann. — Some recent work in the additive theory of numbers 



(n-3°)- 



For every positive integer k, Hardy and Littlewood defined G(k) as the 

 least number s such that every sufficiently large integer is a sum of s &th 

 powers (of positive integers). The object of this paper is to show that, 

 if k ^ 4, then 



(1) G(k) ^ zm + 7 + [2*- 1 (* — 2)(i — fc- I ) w+I ]. 



