Toomas Karmo: Remark in Lieu of Forthcoming Part C of Essay-in-praise-of-Moise

Perpetually at my desk is the obituary, from Toronto's Globe and Mail of 2003-02-07, of  Prof. Max Edelstein (written by his daughter, the British Columbia mathematician Leah Keshet). Prof. Keshet and I have been in correspondence in recent years, and I am inexpressibly grateful to her for the gift of two books from her distinguished father's personal library. - Prof. Edelstein, as my teacher in a special analysis course at Dalhousie University in the 1971 spring semester, conceived the perfect method of teaching. There was no exam. There was no mid-term. There were not even any problem sets, in the normal sense. The sole requirement was that we, the students in his tiny cohort, would write our own textbook, proceeding from hints and questions and challanges handed out by Prof. Edelstein in blue spirit-duplicator coursenotes. My own textbook, which I have kept carefully and have now also sent in photocopy to Dalhousie University, starts with the definition of a field, soon proceeding to a run of theorems. It is just like the first chapter of the wonderful Moise. But I do notice a tiny blemish, common to Prof. Edelstein and the immortal Moise. We really should not say, in postulating the additive identity, 'There exists a unique 0 such that for every element a in the field, 0 + a = a" (and likewise for the mulltiplicative identity). It suffices to say "There exists at least one element 0 such that for every element a in the field, 0 + a = a" (and similarly for the multiplicative identity). Uniqueness is then something we prove, from the field postulates (both in the case of the additive identity and in the case of the multiplicative identity). - The shiny ceramic sphere, with its blue, green, and red triangles, persuades me that in spherical geometry, in contrast with Euclidean plane geometry,  isosceles triangles with a common apex can fail to be similar: the equal base angles of my red isosceles triangle, in particular, on the far side of the sphere, are obtuse, in startling contrast with the equal-and-acute base angles of my blue isosceles triangle, and in startling contrast with the equal-and-acute base angles of my green isosceles triangle. 

Quality assessment: 

On the 5-point scale current in Estonia, and surely in nearby nations, and familiar to observers of the academic arrangements of the late, unlamented, Union of Soviet Socialist Republics (applying the easy and lax standards Kmo deploys in his grubby imaginary "Aleksandr Stepanovitsh Popovi nimeline sangarliku raadio instituut" (the "Alexandr Stepanovitch Popov Institute of Heroic Radio") and his grubby imaginary "Nikolai Ivanovitsh Lobatshevski nimeline sotsalitsliku matemaatika instituut" (the "Nicolai Ivanovich Lobachevsky Institute of Socialist Mathematics") - where, on the lax and easy grading philosophy of the twin Institutes, 1/5 is "epic fail", 2/5 is "failure not so disastrous as to be epic", 3'5 is "mediocre pass", 4.5 is "good", and 5/5 is "excellent"): 4/5. Justification: Kmo had time to do a reasonably complete and (within the framework of the version 1.0.1, 1.0.2, .. process) reasonably polished job. 

Revision history:

• UTC=20230526T0813Z/version 2.1.0: Kmo rectified an embarrassing error. He had written "eight-element" where "sixteen-element" was needed.
• UTC=20160830T0233Z/version 2.0.0: Kmo managed to get rid of the point-form outline and to a proper upload. He reserved the right to do tiny cosmetic, nonsubstantive, tweaks over the coming 48 hours, as here-undocumented versions 2.0.1, 2.0.2, 2.0.3, ... .  
• UTC=20160830T0003Z/version 1.0.0: Kmo lacked  time to be thorough, and so uploaded mere point-form outline. He resolved to get this (admittedly short) blog posting correctly polished by 20160830T0401Z .

[CAUTION: A bug in the blogger software has in some past weeks shown a propensity to insert inappropriate whitespace at some late points in some of my posted essays. If a screen seems to end in empty space, keep scrolling down. The end of the posting is not reached until the usual blogger "Posted by Toomas (Tom) Karmo at" appears.]

Here is a mere public-service announcement. 

I had no time this week for writing up "Part C" of my essay praising Moise, but I hope to write it up next week (with upload in the normal four-hour interval, in this case the four-hour interval UTC=20160906T0001Z/20160906T0401Z). 

In the mean time, readers may wish to follow me in proving an easy general result pertinent to Moise's first chapter, on fields. (The result emerges quickly from thinking about Moise, although Moise does not himself include it in the chapter problem sets.)  Suppose F is any field - finite or infinite. F might, for instance, be the tiny two-element field which Prof. Edelstein drew to the notice of his 1971-spring-semester special Dalhousie University analysis class. Or F might be the rational numbers with standard addition and multiplication. Or F might be the real numbers with standard addition and multiplication. No matter what F is, we can construct a further field, whose elements are pairs (a,b) of F-elements, as follows: addition is defined in terms of F-addition, as (a, b) schplus (c,d) = (a plus c, b plus d), with (0,0) as an (and, we can quickly prove, as the unique) additive identity element; multiplication is defined in terms of F-multiplication, and the additive inverse "minus" of F, as (a, b) schtimes (c, d) = ((a times c) minus (b times d), (a times d) plus (b times c)), and with (1,0) as the identity element. (The task is to prove that the set of such pairs, with schplus and schtimes, is itself a field.)  - So given the reals, we can construct the complex numbers as pairs of reals; and given Prof. Edelstein's tiny two-element field, we can construct a four-element field as pairs of Edelstein-tinies; and given the just-constructed four-element field, we can construct a sixteen-element field; and so on.