By now I am fully established and well under way in Göttingen's university environment and can therefore tell you about my impressions and the things I have dealt with. Here they cultivate, more than anything else, the theory of groups and the so-called modern algebra. The one who deals with the theory of groups is usually Weyl. This year he is teaching differential geometry, but in the library one can see the lecture notes of many previous courses and these are dedicated to the theory of groups. Besides, also this year in the seminar, he covers the theory of groups. Also the courses of professor Herglotz (modular functions and Lie groups) are dedicated to the theory of groups. By the theory of groups one has to understand almost always the theory of finite groups and consequently we slip back, of course, into algebra. Algebra is well known here and Van der Waerden's book is important. Also the things of Emmy Noether take on this spirit. By contrast, in our country, modern algebra has not entered the academic area in the smallest way.
And I confess that I was pretty perplexed during the first lessons; because these German algebraists have a lot of new things and they always use a language full of unknown terms, for me who has never delved into algebra nor into the theory of finite groups. But now I am back on my feet, and I begin to fully understand which is the level at which these people operate. I am studying algebra and making rapid progress. I notice that almost always one stays within the theory of finite groups, while the case of the infinite sets is never touched on. In Van der Waerden's book the subject itself is barely touched on. Certainly the topic is not easy, but it is very likely that combining the ideas we have, for instance the theory of functionals, with the notions that are in common use here, we could find an easy solution. The fact is that this cannot come to anyone's mind, because the theory of functionals is totally unknown.
All the work is done as usual with an algebraic arithmetic spirit. I have been able to sound out a bit the psychology of the students here. They are very unilateral, scientifically speaking. Besides the fact that they do not know about what is not specially fostered in their country, as mathematical physics and the theory of functionals, it happens also that they are not experts in any field, not even in their own. The particular study of Klein's works, for instance, is not a frequent thing here. Of course all the interest that there is here in the theory of groups and for algebra is partly the work of Klein, but Klein as such is already forgotten, and the new algebraists are the ones really known overall. And it is, however, very easy to meet people who do not know anything about automorphic functions or modular elliptic functions and about all of Klein and Poincaré's researches. Of course I speak here about the students of average abiliities.
I began, from the first months of my stay in Germany, the study of "Modern Algebra". This requires me to master a vocabulary and a remarkably complex and intricate system of names, made also difficult by the deliberately private form of each heuristic element in which German mathematicians use to present their writings.
Yet I have done a fairly good practice amid this large number of definitions and concepts, whose distinction is often very subtle. But as I acquired knowledge of the terminology, the fact that the new discipline could have value only from a methodical point of view became more and more evident to me. The intrinsic content instead was simply the content of what has always been known as higher algebra. If we take as an example the case of Galois theory, the results that are to be found in the exposition of "Modern Algebra" do not differ, except in their form, from the ones we find in Bianchi's treatise. What can certainly be stated is that the results added by German mathematicians are out of all proportions regarding the difficulty which arises from the introduction of such a complex phraseology. According to this way of thinking, more critique and governing than constructive, algebra is cultivated in Germany only as an end in itself and is here a sign that eventual new results could impact on research in other fields (I mean for instance the relation between Galois' theory and Picard's, Vessiot for linear differential equations) is set totally aside.
But "Modern Algebra" does not represent a branch of great originality, in another field, which, to tell the truth, is also quite fostered by German mathematicians, where there are real and important problems. I mean to speak of the theory of numbers, a subject not usually fostered in Italy, while in the whole of Germany, and specially in Göttingen, after Gauss and Riemann it represents a tradition. Such a field, however, is in many ways necessarily not algebraic and uses transcendent means so it is closely connected to the theory of functions. Another part of the theory of numbers (and with this in particular the theory of algebraic numbers) has got rid of every transcendent element in the exposition today given in Germany and has come close to "Modern Algebra".