Algebraic Closure

Last time we mentioned the idea of an algebraically closed field. This is a field which has a root to any polynomial we can write down. But what if we’re thinking about a field like \mathbb Q or \mathbb R that isn’t algebraically closed? What do we do then?

Suppose we have a field k. We can add elements to it until it is algebraically closed. This is called the algebraic closure of k. We’ll denote it \overline k. So for instance, \mathbb R is not algebraically closed, because there is no solution to x^2+1=0 in \mathbb R. So we add i=\sqrt{-1}. Adding i gives us the complex numbers, and these are algebraically closed, so \overline{\mathbb R}=\mathbb C

The idea of a closure is an important one in mathematics. There are closures in topology, Galois theory, linear algebra, geometry, etc. They all have a few common properties:

  • Idempotentness: This is an odd word, but it means if you apply it twice, nothing happens. For instance, the algebraic closure of an algebraically closed field is itself. What’s the algebraic closure of \mathbb C? Well, I don’t need to add anything to make it algebraically closed, so it’s itself.
  • The closure, in whatever context, should contain the object you started with. For instance, \mathbb C contains \mathbb R.
  • Closure is monotone: If X is contained in Y, then the closure of X is contained in the closure of Y.
These properties are important mathematical ideas, but if all you cared about was the algebraic closure, they wouldn’t be important at all. This is just to give you some background, if you want it.

Cool. So what about \mathbb Q? Well, \mathbb C is an algebraically closed field containing \mathbb Q, but it’s not the smallest. What is the smallest? Is there a smallest? By now you should be asking these kinds of questions: Does every field have a smallest algebraically closed extension? Does every field even have an algebraically closed extension?

These are reasonable questions, and important ones. To answer these effectively I’ll have to delve farther into ring theory than I am wiling to, so suffice it to say that every field has a unique smallest field that is an extension and algebraically closed. In my next post, I’ll give a “proof” of this fact. I put the “proof” in quotes because the proof relies on ideas in ring theory that, while they aren’t complicated, are annoying and time consuming enough to develop, that I’m going to sweep them under the rug, so to speak.

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