Is there any field of characteristic two that is not $mathbbZ}2 ... | characteristic two
Toabeginner,knowinghowonecouldthinkofananswerisatleastasimportantasknowingananswer.ForexamplesinAlgebra,oneneeds(atleast)twothings:Acatalogueofthebasicstructuresthatappearcommonlyinimportantmathematics,andmethodsofconstructingnewstructuresfromold.Yourcatalogueandconstructionswillnaturallyexpandasyoustudymore,soyoudontneedtoworryaboutthisconsciously.ThemoralIamtryingtoimpartisthefollowing:Insteadoftryingtoconstructaparticularastructurewithparticularproperties"fromscratch"(likeIconstantlytried...
To a beginner, knowing how one could think of an answer is at least as important as knowing an answer.
For examples in Algebra, one needs (at least) two things: A catalogue of the basic structures that appear commonly in important mathematics, and methods of constructing new structures from old. Your catalogue and constructions will naturally expand as you study more, so you dont need to worry about this consciously. The moral I am trying to impart is the following: Instead of trying to construct a particular a structure with particular properties "from scratch" (like I constantly tried to when I was starting to learn these things), first search your basic catalogue. If that doesnt turn up anything, more often than not your search will hint at some basic construction from one of these examples that will work.
When you start learning field theory, your basic catalogue should include all of the finite fields, the rationals, real numbers, complex numbers, and algebraic...