Abstract Algebra - Algebraic Extensions - Pinter Ch27

I titled this "and transcendental extensions" and then they don't appear anywhere in the discussion! So much to verify. A transcendental extension is what you get when you want to extend F to include something that does NOT satisfy an algebraic equation. Transcendental numbers are hard to come by, but they are abundant. Pi is one such example. Luckily for us, we will focus on algebraic extensions. If you want to extend F by a transcendental number, you have to adjoin that number and its inverse, and what you get is basically just "stuff that looks like rational functions in that new element like pi," because no equation is ever satisfied that can simplify it. In that sense, these are somewhat boring. There are other perspectives from which they are more interesting, such as when they appear in geometry, but for now, we content ourselves to say nothing on the matter! Part of the reason this video has to be so long is that I am not talking about rings anywhere. If we had had 2-3 weeks on the theory of rings, most of what I had said would be stuff you already know!