1.4 KiB
Dummit & Foote Chapter 12 - Field Theory
Section 13.1 Basic Theory of Field Extensions
Definition. The charactaristic 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 \amod{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