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Si 
\begin_inset Formula $K$
\end_inset

 es un cuerpo y 
\begin_inset Formula $A$
\end_inset

 es un anillo, todo homomorfismo de anillos 
\begin_inset Formula $h:K\to A$
\end_inset

 es un isomorfismo 
\begin_inset Formula $K\to h(K)$
\end_inset

.
 En efecto, 
\begin_inset Formula $\ker h\subseteq K$
\end_inset

 es un ideal y los únicos ideales de 
\begin_inset Formula $K$
\end_inset

 son 0 y 
\begin_inset Formula $K$
\end_inset

, pero 
\begin_inset Formula $1\notin\ker h$
\end_inset

, luego 
\begin_inset Formula $\ker h=0$
\end_inset

, 
\begin_inset Formula $h$
\end_inset

 es inyectivo y 
\begin_inset Formula $h:K\to h(K)$
\end_inset

 es biyectivo.
\end_layout

\begin_layout Standard
Una 
\series bold
extensión
\series default
 (
\series bold
de cuerpos
\series default
) es un par de cuerpos 
\begin_inset Formula $(K,L)$
\end_inset

 con 
\begin_inset Formula $K$
\end_inset

 es subcuerpo de 
\begin_inset Formula $L$
\end_inset

, que representamos como 
\begin_inset Formula $K\subset L$
\end_inset

, 
\begin_inset Formula $L/K$
\end_inset

 o, si 
\begin_inset Formula $K\neq L$
\end_inset

, como 
\begin_inset Formula $K\subsetneq L$
\end_inset

.
\end_layout

\begin_layout Standard
Algunas extensiones son 
\begin_inset Formula $\mathbb{Q}\subset\mathbb{R}$
\end_inset

, 
\begin_inset Formula $\mathbb{R}\subset\mathbb{C}$
\end_inset

 y, para todo cuerpo 
\begin_inset Formula $K$
\end_inset

, 
\begin_inset Formula $K\subset K(X)$
\end_inset

, donde 
\begin_inset Formula $K(X)$
\end_inset

 es el cuerpo de fracciones de 
\begin_inset Formula $K[X]$
\end_inset

.
 Otras son 
\begin_inset Formula $\mathbb{Q}\subset\mathbb{Q}[\sqrt{m}]$
\end_inset

 para 
\begin_inset Formula $m\in\mathbb{Z}$
\end_inset

 no cuadrado y 
\begin_inset Formula $\mathbb{Q}\subset\mathbb{Q}[i]$
\end_inset

.
 En ef
\begin_inset Note Note
status open

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\begin_inset ERT
status open

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\end_inset


\end_layout

\begin_layout Standard
Sea 
\begin_inset Formula $K$
\end_inset

 un cuerpo no trivial, existe un subcuerpo 
\begin_inset Formula $K'$
\end_inset

 de 
\begin_inset Formula $K$
\end_inset

 llamado 
\series bold
subcuerpo primo
\series default
 de 
\begin_inset Formula $K$
\end_inset

 contenido en cualquier subcuerpo de 
\begin_inset Formula $K$
\end_inset

, y este es isomorfo a 
\begin_inset Formula $\mathbb{Z}_{p}$
\end_inset

 si la característica de 
\begin_inset Formula $K$
\end_inset

 es un entero primo 
\begin_inset Formula $p$
\end_inset

 o a 
\begin_inset Formula $\mathbb{Q}$
\end_inset

 en caso contrario.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
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\end_inset


\end_layout

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