9 February - 15 February

We reviewed some basic structures and theorems that will be needed throughout the course. In particular, we recalled the notions of ring, domain, field, ideal of a domain, principal ideal domain (PID), prime element and prime ideal, irreducible element and maximal ideal.

We proved Eisenstein's irreducibility criterion and saw Gauss' Lemma on the irreducibility of polynomials over unique factorization domains and their field of fractions.

We saw that a field K is a vector space over any subfield F, and defined degree of the extension K/F to be the dimension of K as an F-space. We denote it by $[K : F]$. We saw examples.

If $K/F$ is a field extension and $X \subseteq K$, we defined the extension of $F$ that is generated by $X$ over $F$, and denoted is by $F(X)$. We saw $F(X)$ for finite sets $X$.

We defined the notion of a algebraic element $\alpha$ over $F$, and of an algebraic extension $K/F$. For $\alpha$ algebraic over $F$, we defined the minimal polynomial $\min(F, \alpha)$ and proved some basic facts.

We studied the simple extension $F(\alpha)/F$ for $\alpha$ algebraic over $F$.

Reading: pages 1 - 5 from the notes.