3
$\begingroup$

I have come across the following reference to D K Faddeev's construction of quaternionic fields in the book The Embedding Problem in Galois Theory by Ishkhanov, Lur'e and Faddeev :

[45] D. K. Faddeev, Construction of algebraic domains whose Galois group is the quaternionic group, Leningrad. Gos. Univ. Uchen. Zap. 3 (1937), no. 17, 17--23,

which is presumably the same as an item in the bibliography of Faddeev's survey article ТЕОРИЯ ГАЛУА (В МИАНе) :

[16] Фаддеев Д. К. Построение алгебраических областей, группой Галуа которых является группа кватернионов. — Учен. зап. ЛГУ, 1936, т. 17, с. 17—25.

Edit KConrad has kindly provided a link to the English translation of this survey (Спасибо, Кит). Faddeev says

In a paper of mine in 1937 [16], the problem of constructing fields with quaternionic groups over $\mathbf{Q}$ was solved. The algebraic part of the construction can be extended to any field of characteristic different from $2$ which has a sufficient number of quadratic extensions. The arithmetic part allows one to give an algorithm for construction of fields over $\mathbf{Q}$ in the order of growth of their discriminants.

I couldn't find this paper at mathnet.ru, nor is it listed in the Zentralblatt. There is a later paper with a similar title

Construction of fields of algebraic numbers whose Galois group is a group of quaternion units, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 47, 390-392 (1945)

which is listed but not reviewed in the Zentralblatt; it is not be found at mathnet.ru either.

Question. What does Faddeev prove in these papers ?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
5
$\begingroup$

I don't have the paper you refer to, but on the page http://www.math.spbu.ru/vestnik/2008/vestnik0801/dfaddeev.pdf, which is dedicated to the 100th birthday of Faddeev, his early work is described in the following way (3rd paragraph of the second page):

"The second direction which interested Faddeev in the first years of his scientific activity was Galois theory. He was especially interested in the so-called inverse Galois problem (which to this day is still not solved): construct an extension of a given base field with a prescribed Galois group. The first results of Faddeev in this area were related to the construction of extensions with small Galois groups: subgroups of $S_4$, metacyclic transitive groups of prime degree, and groups of quaternions and quaternionic units. He used a beautiful geometric approach: the sought-after field was interpreted as a subset of a vector space on which the Galois group acts in a rather simple way. Many of the results had an elegant geometric formulation. For instance, an extension of the rationals with quaternionic Galois group is closely related to triples of pairwise orthogonal vectors in ${\mathbf R}^3$ with rational coordinates."

The article then goes on to discuss later work.

Improve this answer
$\endgroup$
6
  • 2
    $\begingroup$ I too am intrigued by what his geometric construction could be, since I am teaching Galois theory at the moment and such a construction of an extension of ${\mathbf Q}$ with quaternionic Galois group would be great to show the class (better than presenting explicit field generators out of nowhere). If someone has easy access to the papers, could they indicate what the construction in them is? $\endgroup$
    – KConrad
    Commented Apr 16, 2013 at 12:15
  • $\begingroup$ Hmm, I noticed that the beginning of the article I cited is very similar to an article on Faddeev for his 80th birthday (already written in English): mathsoc.spb.ru/pantheon/faddeev/UMN-89e.html. The only difference in the part I wrote above is that the one written for his 80th birthday includes subgroups of $S_3$ in the list before subgroups of $S_4$. $\endgroup$
    – KConrad
    Commented Apr 16, 2013 at 12:31
  • $\begingroup$ One more reference: in a survey on Galois theory at the Steklov Institute (books.google.com/…) Faddeev writes at the end of section 5 that the algebraic part of his construction of quaternionic Galois extensions extends to any base field outside characteristic 2 with enough quadratic extensions. $\endgroup$
    – KConrad
    Commented Apr 16, 2013 at 12:47
  • $\begingroup$ I got curious about these papers of Faddeev because Witt (Crelle, 1936), Reichardt (Zeitschrift, 1936) and Richter (Annalen, 1935) gave constructions of quaternionic extensions around the same time, and whereas at least the first two authors are mentioned in later papers by Western mathematicians, the only reference to Faddeev I've seen is in the Russian book on The embedding problem. $\endgroup$
    – Chandan Singh Dalawat
    Commented Apr 16, 2013 at 12:59
  • $\begingroup$ The Steklov Institute paper looks like the English translation of the survey paper I mentioned in the question. Thank you very much for providing the link; it will save me a lot of calls to translate.google.com. $\endgroup$
    – Chandan Singh Dalawat
    Commented Apr 16, 2013 at 13:11

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .