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.

As mentioned, the intended audience is high school or early college students who are already familiar with complex numbers, and want to get a glimpse of the “big picture” that they illustrate, at least from an algebraic standpoint. It will be of particular interest to those students who are pursuing (or intend to pursue) a major in mathematics but have not yet had a course in abstract algebra. The transition from largely computation-based courses in calculus to concept-based courses in abstract algebra is notoriously a difficult one for a lot of students, and having been introduced to at least some of the concepts beforehand will be of great help.

Except for the first post which will introduce basic definitions and notations, I plan to describe each concept in a separate post, and thus posts will be fairly short. Excessive generalisation will be avoided, and in particular all concepts wil be illustrated by examples taking place in the extension $\mathbf{C}/\mathbf{R}$. Also, since the goal is to introduce concepts, not to give a rigorous treatment, almost no result will be proven.

At the end of each post will be a number of questions that you can use to test your understanding of the concepts, those questions will sometimes involve other fields than $\mathbf{C}$ and $\mathbf{R}$. They are purely computational or conceptual, and do not require elaborate proofs. Of course this is not school, you do not have to turn them in. ;) As above, though, the aspiring mathematician would be well-advised to at least give them some thought. I am of course always available to answer questions or check your answers if you have doubts.

Those in the know will probably notice that my exposition will be closely following that of Fraleigh’s text, from which I learned the subject myself and which I warmly recommend to the interested reader.

The posts are as follows:

0 – Notation and definitions: fields and polynomials
1 – Roots of a polynomial
2 – Irreducible polynomials
3 – Kronecker’s theorem
4 – Algebraic elements
5 – Simple extensions
6 – Simple algebraic extensions
7 – Summary of the construction of C

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