Some time ago, I discovered what seemed like a basic new result about the real numbers. The reason I thought it was new was (a) it was basic and had a fairly easy proof, (b) it was not mentioned in the usual "foundations of real analysis" books but due to (a) it should have been. However, I now suspect it is not new, the problem is merely that those books are crap. Anyhow, here are some of "my" results: 1. I can explicitly construct a dense subfield F of the real numbers, having the same cardinality as the real numbers, but omitting all algebraic irrationals. 2. I also can explicitly construct a continuum infinity of such subfields, call them F(x) where x, 0<x<1, is a real parameter, such that the intersection of any two of them is the rational numbers, i.e. they are "almost disjoint" -- and if any finite set of irrationals is chosen, each element from a different F(x), then that set will automatically be algebraically independent. [By "explicitly" I mean (among other things) that I do not require the axiom of choice.] 3. All those subfields each have measure=0. It is impossible for any additive subgroup of the reals to exist, which has positive measure. (The proof of this impossibility is mostly stolen from a recent mathfun post by Fred W. Helenius.) 4. Results similar to the above also work inside the complex numbers, quaternions, octonions, and "16ons." All of these subfields are disconnected. Some literature references, which I have not seen, which apparently prove similar statements, include: Jean Dieudonne: Sur les corps topologiques connexes, Comptes Rendus Acad. Sci. Paris 221 (1945) 396-398. Carsten Elsner: Uber eine effektive Konstruktion grosser Mengen algebraisch unabhangiger Zahlen, Mathematische Semesterberichte 47 (2000) 243-256. Isaac Kapuano: Sur les corps de nombres a une dimension distincts du corps reel, Rev. Fac. Sci. Univ. Istanbul (A) 11 (1946) 30-39. Hellmuth Kneser: Eine Kontinuumsmachtige, algebraisch unabhangige Menge reeler Zahlen, Bull.Soc.Math.Belg. 12 (1960) 23-27. Ernst Steinitz: Algebraische Theorie der Körper, Journal fur.Math 137 (1910) 167-309. Dieudonne and/or Kapuano actually apparently have shown existence of a CONNECTED (indeed, more strongly, "locally connected") dense subfield of the complex numbers (but not "arc connected") including one that is isomorphic with the real numbers. But I think in order to accomplish that they needed to use the Axiom of Choice so their proof of existence is highly nonconstructive. [It is impossible to make a connected additive subgroup of the reals, which is why they were doing complex numbers.] Questions: 1. Do they really require the Axiom of Choice? 2. If the rationals are removed from this connected subfield, does that disconnect it?