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-rw-r--r--ga/n2.lyx532
1 files changed, 368 insertions, 164 deletions
diff --git a/ga/n2.lyx b/ga/n2.lyx
index 4216f33..6bdbc01 100644
--- a/ga/n2.lyx
+++ b/ga/n2.lyx
@@ -130,9 +130,9 @@ subcuerpo
\end_inset
es inyectivo.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
\begin_layout Description
\begin_inset Formula $[1\implies2]$
\end_inset
@@ -164,6 +164,7 @@ subcuerpo
.
\end_layout
+\begin_deeper
\begin_layout Description
\begin_inset Formula $[2\implies1]$
\end_inset
@@ -246,6 +247,11 @@ subcuerpo
\end_layout
\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Un elemento
\begin_inset Formula $a\in A$
@@ -256,9 +262,9 @@ Un elemento
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open
@@ -279,6 +285,7 @@ status open
.
\end_layout
+\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open
@@ -320,6 +327,11 @@ Si
\end_layout
\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset CommandInset label
LatexCommand label
@@ -336,24 +348,28 @@ name "enu:char-domain"
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Consecuencia de lo anterior.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Todo subanillo de un dominio es un dominio.
\end_layout
\begin_layout Enumerate
La característica de un dominio no trivial es 0 o un número primo.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Si la característica de
\begin_inset Formula $A$
\end_inset
@@ -412,9 +428,6 @@ noprefix "false"
\end_layout
-\end_deeper
-\begin_layout Standard
-\begin_inset Newpage pagebreak
\end_inset
@@ -457,9 +470,9 @@ Para
\end_inset
es primo.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
\begin_layout Description
\begin_inset Formula $[1\implies3]$
\end_inset
@@ -475,6 +488,7 @@ Para
.
\end_layout
+\begin_deeper
\begin_layout Description
\begin_inset Formula $[3\implies2]$
\end_inset
@@ -522,6 +536,11 @@ Para
\end_layout
\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Si
\begin_inset Formula $m\in\mathbb{Z}$
@@ -536,10 +555,10 @@ Si
\end_inset
es cuerpo.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Ambos son subanillos de
\begin_inset Formula $\mathbb{C}$
\end_inset
@@ -617,20 +636,28 @@ Ambos son subanillos de
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Un producto de anillos no triviales nunca es un dominio.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $(1,0)(0,1)=(0,0)$
\end_inset
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Section
Ideales maximales y primos
\end_layout
@@ -688,9 +715,9 @@ name "enu:char-maximal"
\end_inset
es un cuerpo.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open
@@ -747,6 +774,7 @@ Si
\end_layout
+\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open
@@ -792,6 +820,11 @@ Los únicos ideales de
\end_layout
\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset CommandInset label
LatexCommand label
@@ -808,9 +841,9 @@ name "enu:char-prime"
\end_inset
es un dominio.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open
@@ -847,6 +880,7 @@ Sean
es un dominio.
\end_layout
+\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open
@@ -892,16 +926,21 @@ Sean
\end_layout
\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Si
\begin_inset Formula $I$
\end_inset
es maximal, es primo.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Si es maximal,
\begin_inset Formula $A/I$
\end_inset
@@ -913,16 +952,20 @@ Si es maximal,
es primo.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $A$
\end_inset
es cuerpo si y sólo si 0 es maximal.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $A\cong A/0$
\end_inset
@@ -939,16 +982,20 @@ noprefix "false"
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $A$
\end_inset
es dominio si y sólo si 0 es primo.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $A\cong A/0$
\end_inset
@@ -965,7 +1012,11 @@ noprefix "false"
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Dado un conjunto
\begin_inset Formula $S$
@@ -998,7 +1049,11 @@ lema de Zorn:
\begin_layout Standard
Todo ideal propio de un anillo está contenido en un ideal maximal.
-
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
\series bold
Demostración:
\series default
@@ -1096,6 +1151,11 @@ Demostración:
.
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Section
Divisibilidad
\end_layout
@@ -1229,10 +1289,10 @@ Si
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Sea
\begin_inset Formula $x$
\end_inset
@@ -1252,7 +1312,11 @@ Sea
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Dos elementos
\begin_inset Formula $a,b\in A$
@@ -1280,22 +1344,6 @@ asociados
\end_layout
\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-begin{samepage}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
Si
\begin_inset Formula $D$
\end_inset
@@ -1321,7 +1369,8 @@ Si
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
\begin_layout Itemize
\begin_inset Argument item:1
@@ -1416,17 +1465,6 @@ Claramente
.
\end_layout
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-end{samepage}
-\end_layout
-
\end_inset
@@ -1478,7 +1516,11 @@ Si
\end_inset
es un dominio, todo primo es irreducible.
-
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
\series bold
Demostración:
\series default
@@ -1555,9 +1597,18 @@ Demostración:
es análogo.
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Irreducible en un dominio no implica primo.
-
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
\series bold
Demostración:
\series default
@@ -1648,6 +1699,11 @@ Demostración:
.
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Sean
\begin_inset Formula $A$
@@ -1679,9 +1735,9 @@ Sean
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
\begin_layout Description
\begin_inset Formula $[1\implies2]$
\end_inset
@@ -1713,6 +1769,7 @@ Sean
.
\end_layout
+\begin_deeper
\begin_layout Description
\begin_inset Formula $[2\implies3\implies1]$
\end_inset
@@ -1721,6 +1778,11 @@ Sean
\end_layout
\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $a$
\end_inset
@@ -1749,10 +1811,10 @@ Sean
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $a$
\end_inset
@@ -1799,7 +1861,11 @@ traduciendo
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Si
\begin_inset Formula $A$
@@ -1826,9 +1892,9 @@ Si
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open
@@ -1894,6 +1960,7 @@ Como
.
\end_layout
+\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open
@@ -1977,6 +2044,11 @@ Como
\end_layout
\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Dados un anillo conmutativo
\begin_inset Formula $A$
@@ -2064,10 +2136,10 @@ mínimo común múltiplo
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $a$
\end_inset
@@ -2104,7 +2176,11 @@ mínimo común múltiplo
Juntando ambos se obtiene el resultado.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $a=\text{mcm}S$
\end_inset
@@ -2131,10 +2207,10 @@ mínimo común múltiplo
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $a$
\end_inset
@@ -2167,7 +2243,11 @@ mínimo común múltiplo
Juntando ambos se obtiene el resultado.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Si
\begin_inset Formula $a=\text{mcd}S$
@@ -2190,15 +2270,19 @@ Si
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Se obtiene de la caracterización de máximo común divisor y la de asociados
por ideales principales.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Si
\begin_inset Formula $a=\text{mcm}S$
@@ -2221,14 +2305,18 @@ Si
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Análogo.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset CommandInset label
LatexCommand label
@@ -2274,10 +2362,10 @@ identidad de Bézout
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Si
\begin_inset Formula $b\in A$
\end_inset
@@ -2293,7 +2381,11 @@ Si
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $\text{mcd}S=1$
\end_inset
@@ -2307,9 +2399,9 @@ Si
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open
@@ -2339,6 +2431,7 @@ Si hubiera un divisor común
.
\end_layout
+\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open
@@ -2360,6 +2453,11 @@ status open
\end_layout
\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Si
\begin_inset Formula $1\in(S)$
@@ -2370,10 +2468,10 @@ Si
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Basta aplicar
\begin_inset CommandInset ref
LatexCommand ref
@@ -2387,7 +2485,11 @@ noprefix "false"
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Section
Dominios de factorización única
\end_layout
@@ -2534,10 +2636,10 @@ Dado
\end_inset
es un DF.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Si
\begin_inset Formula $m$
\end_inset
@@ -2620,6 +2722,7 @@ Si
.
\end_layout
+\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open
@@ -2652,7 +2755,7 @@ Si
.
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
Sea ahora
\begin_inset Formula $x=a+b\sqrt{m}\neq0$
\end_inset
@@ -2753,6 +2856,11 @@ Sea ahora
\end_layout
\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Un dominio
\begin_inset Formula $D$
@@ -3053,13 +3161,6 @@ Un dominio
Por tanto las factorizaciones iniciales son equivalentes.
\end_layout
-\begin_layout Standard
-\begin_inset Newpage pagebreak
-\end_inset
-
-
-\end_layout
-
\begin_layout Section
Dominios de ideales principales
\end_layout
@@ -3324,10 +3425,10 @@ El cuadrado del módulo complejo es una función euclídea en
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Si
\begin_inset Formula $x:=a+bi$
\end_inset
@@ -3415,7 +3516,11 @@ Si
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Sean
\begin_inset Formula $\delta$
@@ -3442,7 +3547,8 @@ Sean
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
\begin_layout Itemize
\begin_inset Argument item:1
@@ -3582,12 +3688,22 @@ Sea
\end_layout
\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Como
\series bold
teorema
\series default
-, todo dominio euclídeo es DIP, pues si
+, todo dominio euclídeo es DIP
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+, pues si
\begin_inset Formula $\delta$
\end_inset
@@ -3611,6 +3727,11 @@ teorema
\begin_inset Formula $I=(a)$
\end_inset
+
+\end_layout
+
+\end_inset
+
.
\end_layout
@@ -3636,7 +3757,8 @@ Si
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
\begin_layout Description
\begin_inset Formula $[1\iff3]$
@@ -3683,6 +3805,11 @@ Si
Obvio.
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Section
Cuerpos de fracciones
\end_layout
@@ -3747,7 +3874,11 @@ Llamamos
\end_inset
están bien definidas.
-
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
\series bold
Demostración:
\series default
@@ -3788,6 +3919,11 @@ Demostración:
.
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Propiedades:
\begin_inset Formula $\forall a,b\in D;s,t\in D\setminus\{0\}$
@@ -3801,81 +3937,101 @@ Propiedades:
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $\frac{a}{s}=\frac{0}{1}\iff a=a1=s0=0$
\end_inset
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $\frac{a}{s}=\frac{1}{1}\iff a=s$
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $\frac{a}{s}=\frac{1}{1}\iff a=a1=s1=s$
\end_inset
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $\frac{at}{st}=\frac{a}{s}$
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $ats=ast$
\end_inset
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $\frac{a}{s}=\frac{b}{s}\iff a=b$
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $\frac{a}{s}=\frac{b}{s}\iff as=bs\overset{s\neq0}{\iff}a=b$
\end_inset
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $\frac{a}{s}+\frac{b}{s}=\frac{a+b}{s}$
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $\frac{a}{s}+\frac{b}{s}=\frac{as+bs}{ss}=\frac{(a+b)s}{ss}=\frac{a+b}{s}$
\end_inset
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
De aquí,
\begin_inset Formula $(Q(D),+,\cdot)$
@@ -3902,7 +4058,11 @@ de cocientes
\end_inset
.
-
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
\series bold
Demostración:
\end_layout
@@ -3970,6 +4130,11 @@ Demostración:
.
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Así,
\begin_inset Formula $\mathbb{Q}$
@@ -4049,10 +4214,10 @@ Propiedad universal del cuerpo de fracciones:
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Si
\begin_inset Formula $\hat{f}:Q(D)\to K$
\end_inset
@@ -4102,7 +4267,11 @@ Si
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Sean
\begin_inset Formula $K$
@@ -4121,10 +4290,10 @@ Sean
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
El homomorfismo
\begin_inset Formula $f:=g\circ u=h\circ u$
\end_inset
@@ -4152,7 +4321,11 @@ El homomorfismo
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Sean
\begin_inset Formula $F$
@@ -4187,10 +4360,10 @@ Sean
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Por la Propiedad Universal de
\begin_inset Formula $Q(D)$
\end_inset
@@ -4236,7 +4409,11 @@ Por la Propiedad Universal de
triviales, es el isomorfismo buscado.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Sean
\begin_inset Formula $D$
@@ -4259,7 +4436,11 @@ Sean
\end_inset
.
- En efecto, por la propiedad universal, existe un homomorfismo
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+En efecto, por la propiedad universal, existe un homomorfismo
\begin_inset Formula $\tilde{f}:Q(D)\to K$
\end_inset
@@ -4278,6 +4459,11 @@ Sean
.
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
De aquí, para
\begin_inset Formula $m\in\mathbb{Z}$
@@ -4296,7 +4482,11 @@ De aquí, para
\end_inset
.
-
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
\series bold
Demostración:
\series default
@@ -4357,6 +4547,11 @@ y recíprocamente,
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Sea
\begin_inset Formula $K$
@@ -4399,7 +4594,11 @@ subcuerpo primo
\end_inset
en caso contrario.
-
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
\series bold
Demostración:
\series default
@@ -4491,5 +4690,10 @@ Demostración:
\end_layout
+\end_inset
+
+
+\end_layout
+
\end_body
\end_document
generated by cgit v1.2.3 (git 2.46.0) at 2025-10-27 08:57:17 +0000