Math 81/111

Algebra II — Rings and Fields
Last updated June 20, 2019 09:46:48 EDT

Course Objectives: Students in high school learn that every polynomial with real coefficients factors completely over the complex numbers, so in particular, every polynomial with rational coefficients also has all its roots in the complex numbers. But $\mathbb C$ is such a large field we can gather little information about the roots of the polynomial.

Suppose instead we let $K$ be the smallest subfield of $\mathbb C$ containing all the roots of a given polynomial $f \in \mathbb Q[x]$. What properties of $K$ can we deduce and what do those properties tell us about $f$? In the beginning of the course, we investigate how to construct such a field $K$, and observe simple properties. One obvious property is that since $\mathbb Q \subseteq K$ are both fields, we can view $K$ as a vector space over $\mathbb Q$. What natural questions arise? What is its dimension and can we describe a basis? It turns out that $K/\mathbb Q$ has finite dimension $d \le n!$ where $n = \deg(f)$, and that there are several interesting bases.

To go further (into Galois theory) we shall associate a group $G$ to the field extension $K/\mathbb Q$; in particular $G$ is the group of field automorphisms of $K$. The beauty of Galois theory is a delightful correspondence between subgroups of $G$ and fields $E$ with $\mathbb Q \subseteq E \subseteq K$. And properties of the group (cyclic, abelian, solvable, etc) can actually tell us whether all the roots of our given polynomial can be expressed algebraically in terms of radicals (such as the roots given by the quadratic formula). We discover in the end that unlike polynomials of degree 2, 3, and 4, there can be no formula in terms of radicals which can characterize the roots of every polynomial of degree 5 or greater.