Dummit & Foote Chapter 13 - Field Theory

Section 13.1 Basic Theory of Field Extensions

Definition. The characteristic of a field \(F\) is the smallest positive integer \(p\) such that \(1_F * p = 0\). It follows that \(p\) is \(0\) or prime, and \(p \alpha = 0\) for any \(\alpha \in F\).

Definition. If \(K, F\) are fields such that \(F \subseteq K\), then \(K\) is an extension field or extension of \(F\), denoted \(K / F\).

Definition. The degree (or relative degree or index) of \(K/F\), denoted \([K:F]\), is the dimension of \(K\) as a \(F\)-vector space.

Theorem. Let \(F\) be a field, \(p(x) \in F[x]\). Then, there exists a \(K\) such that \(p(x)\) has a root in \(K\).

Theorem. Let \(F\) be a field, \(p(x) \in F[x]\). Then, \(K = \frac{F[x]}{(p(x))}\) and \(\theta = x a \mod{p(x)}\), \(K\) has a basis of \(1, \theta, \ldots, \theta^{n-1}\) where \(n = \deg(p)\).

Theorem. Let \(K/F\) and \(\alpha, \beta, \ldots \in K\). Then, the smallest subfield of \(K\) containing \(F\) and \(\alpha, \beta, \ldots\) is \(F(\alpha, \beta, \ldots)\), which is the field generated by \(\alpha, \beta, \ldots\) over \(F\).

Definition. If \(K\) is generated by \(F(\alpha)\), then \(K\) is a simple extension of \(F\).

Theorem. Let \(F\) be a field, \(p(x) \in F[x]\) be irreducible. Then, if \(\alpha\) is a root of \(p(x)\) and \(K\) is an extension of \(F\) containing \(\alpha\), then \(F(\alpha) \cong \frac{F[x]}{(p(x))}\).

TODO