# Field theory for high-schoolers

Note: This is something I posted on my Tumblr a while ago. In addition to the purpose expressed below, I am reposting it here so that interested readers even with only a modest background in mathematics will at least have some idea of what I am talking about when I will post some more advanced things.

Field theory is the study of fields; that is, sets of numbers that behave like the usual rational, real, or complex numbers. This is a series of posts that aim to introduce the basic concepts of field theory using examples from number systems the typical high school or early college student is familiar with: the aforementioned real and complex numbers. (Quite surprisingly, even though rational numbers are usually considered simpler than real or complex numbers, studying them from a field-theoretic standpoint is a lot more complicated, and we will leave them largely aside.) In particular, the concept of field extensions (fields contained in larger fields) is central in field theory, and the extension $\mathbf{C}/\mathbf{R}$ provides an example of a field extension which can be used to introduce field-theoretic concepts in a familiar setting.