Field Extensions

Subjects: Field Theory
Links: Rings and Fields, Characteristic of a Ring, Ring Homomorphisms

Prop: Let $\phi :F\to K$ be a homomorphism of fields. Then $\phi$ is either identically $0$ or is injective, so that image is either $0$ or isomorphic to $F$.

Def: A field extension is a field monomorphism $\varphi :F\to K$.

We see that if there's a monomorphism $\varphi :F\to K$, then $F$ is embedded in $K$, so we can treat it as if it was a subset of $K$. Additionally, if $F\subseteq K$, then the monomorphism is the injection, thus it is also a field extension.

Def: If $F$ and $K$ are fields such that $F\subseteq K$ and the operations $(+,\cdot )$ of $F$ are the sames as the ones in $K$, then we say that $F$ is a subfield of $K$ or, equivalently, $K$ is an extension of $F$.

If $F\subseteq K$ is a field extension, we also use the notation $K/F$, $K:F$, and

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Def: The degree, or relative degree, or index of a field extension $K/F$, denoted $[K:F]$, is the dimension of $K$ as a vector space over $F$, i.e., $[K:F]:={\text{dim}}_{F}K$. The extension is said to be finite if $[K:F]$ is finite and is said to be infinite otherwise.

Th: If $L/K$ and $K/F$ are finite field extensions, then $L/F$ is a finite field extension, and $[L:F]=[L:K][K:F]$.

Cor: If $L/F$ is a finite extension, and $K$ is an intermediate field, i.e., $F\subseteq K\subseteq L$, then $[K:F]$ divides $[L:F]$.

Obs: We see that a field is in particular an integral domain, thus their characteristic is either prime or $0$.

Def: Let $F$ be a field, and $\{L_{\alpha }\mid \alpha <\kappa \}$ be the set of all subfields of $F$. We define the prime subfield of $F$ to be $$\bigcap_{\alpha < \kappa} L_\alpha.$$
Prop: Let $F$ be field with prime subfield $K$. The following statements are true.

• If $F$ has characteristic $0$, then $K\cong \mathbb{Q}$.
• If $F$ has nonzero characteristic, then $K\cong \mathbb{Z}/p\mathbb{Z}$.

Def: Let $K/F$ be a field extension and a subset $X\subseteq K$. Let $\mathcal{F}:=\{L\subseteq K\mid L/F\text{ is a field extension and }X\subseteq L\}$. Note that $\mathcal{F}\ne \varnothing$, since $X\in \mathcal{F}$. The subfield $F(X):=\bigcap \mathcal{F}$, and is called the field obtained by adjoining $X$ to $F$.

We see that $F\subseteq F(X)\subseteq K$, and $F(X)$ is the smallest field of $K$ that contain $F$ and $X$. If $X=\{a_1,\dots ,a_n\}\subseteq K$ is finite set, then we use the notation $$F(X) = F(a_1,\dots, a_n). $$In particular, if $X=\{a\}\subseteq K$, we say that is a simple extension $F(a)/F$.

Prop: Let $K/F$ a field extension and $\alpha \in K$. If $F(\alpha )/F$ the simple extension obtained by adjoining $\alpha$ to $F$. Then $$F(\alpha) = \left{\left.\frac{f(\alpha)}{g(\alpha)}; \right\rvert; f(x), g(x)\in F[x], g(\alpha) \neq 0\right} $$