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-rw-r--r--ga/n3.lyx476
1 files changed, 317 insertions, 159 deletions
diff --git a/ga/n3.lyx b/ga/n3.lyx
index 772b2a5..f592c64 100644
--- a/ga/n3.lyx
+++ b/ga/n3.lyx
@@ -269,10 +269,10 @@ Si
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Sean
\begin_inset Formula $P=:\sum_{k}a_{k}X^{k}$
\end_inset
@@ -353,6 +353,7 @@ Si la desigualdad es estricta,
.
\end_layout
+\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open
@@ -382,6 +383,11 @@ El coeficiente de grado
\end_layout
\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $\text{gr}(PQ)\leq\text{gr}(P)+\text{gr}(Q)$
\end_inset
@@ -391,10 +397,10 @@ El coeficiente de grado
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Para
\begin_inset Formula $N>n+m$
\end_inset
@@ -418,7 +424,7 @@ Para
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
El coeficiente de grado
\begin_inset Formula $n+m$
\end_inset
@@ -434,7 +440,11 @@ El coeficiente de grado
, luego la igualdad se da si y sólo si esto es no nulo.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
\begin_inset Formula $A[X]$
\end_inset
@@ -461,7 +471,8 @@ cuerpo de las funciones racionales
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
\begin_layout Enumerate
\begin_inset Argument item:1
@@ -523,6 +534,11 @@ Sean
tampoco lo es.
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Section
Propiedad universal
\end_layout
@@ -581,10 +597,10 @@ PUAP
\end_inset
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Si
\begin_inset Formula $\tilde{f}$
\end_inset
@@ -620,7 +636,11 @@ lo que prueba la unicidad.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $A[X]$
\end_inset
@@ -671,10 +691,10 @@ lo que prueba la unicidad.
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Tomando
\begin_inset Formula $v$
\end_inset
@@ -740,7 +760,11 @@ Tomando
es el isomorfismo buscado.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Así:
\end_layout
@@ -808,17 +832,21 @@ función polinómica
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $S_{b}$
\end_inset
se obtiene al aplicar la PUAP a la inclusión.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Dado
\begin_inset Formula $a\in A$
@@ -837,10 +865,10 @@ Dado
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $S_{X-a}(S_{X+a}(X))=S_{X-a}(X+a)=X$
\end_inset
@@ -872,7 +900,11 @@ Dado
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Si
\begin_inset Formula $A$
@@ -883,10 +915,10 @@ Si
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
El homomorfismo
\begin_inset Formula $A[X]\to A$
\end_inset
@@ -898,7 +930,11 @@ El homomorfismo
, y basta aplicar el primer teorema de isomorfía.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Todo homomorfismo de anillos
\begin_inset Formula $f:A\to B$
@@ -921,10 +957,10 @@ que es inyectivo o suprayectivo si lo es
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Se obtiene de aplicar la PUAP a la composición de la inclusión
\begin_inset Formula $B\to B[X]$
\end_inset
@@ -936,7 +972,11 @@ Se obtiene de aplicar la PUAP a la composición de la inclusión
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Si
\begin_inset Formula $A$
@@ -955,14 +995,18 @@ Si
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Basta aplicar lo anterior al homomorfismo inyectivo inclusión.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Si
\begin_inset Formula $I$
@@ -1001,10 +1045,10 @@ Su núcleo es
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Se obtiene de aplicar la PUAP a la proyección
\begin_inset Formula $A\to A/I$
\end_inset
@@ -1017,7 +1061,11 @@ Se obtiene de aplicar la PUAP a la proyección
es un ideal, y entonces basta aplicar el primer teorema de isomorfía.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Section
Raíces de polinomios
\end_layout
@@ -1078,7 +1126,11 @@ noprefix "false"
\end_inset
.
-
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
\series bold
Demostración:
\series default
@@ -1205,6 +1257,11 @@ Demostración:
.
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
\begin_inset Float algorithm
wide false
@@ -1368,7 +1425,11 @@ Teorema del resto:
\end_inset
.
- En efecto, si
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+En efecto, si
\begin_inset Formula $f=q(X-a)+r$
\end_inset
@@ -1389,6 +1450,10 @@ Teorema del resto:
\end_inset
.
+\end_layout
+
+\end_inset
+
De aquí se obtiene el
\series bold
teorema de Ruffini
@@ -1528,7 +1593,11 @@ La multiplicidad de
\end_inset
no es raíz.
-
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
\series bold
Demostración:
\series default
@@ -1601,6 +1670,11 @@ Demostración:
.
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Si
\begin_inset Formula $D$
@@ -1651,7 +1725,11 @@ Si
\end_inset
.
-
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
\series bold
Demostración:
\series default
@@ -1725,6 +1803,11 @@ Demostración:
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
\series bold
@@ -1767,10 +1850,10 @@ Para
\end_inset
son iguales.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Sean
\begin_inset Formula $a_{1},\dots,a_{m}$
\end_inset
@@ -1790,7 +1873,11 @@ Sean
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $D$
\end_inset
@@ -1804,9 +1891,9 @@ Sean
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open
@@ -1832,6 +1919,7 @@ Si hubiera
.
\end_layout
+\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open
@@ -1854,6 +1942,11 @@ Si
\end_layout
\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Como ejemplo de lo anterior, por el teorema pequeño de Fermat, dado un primo
@@ -1913,10 +2006,10 @@ derivada
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula
\begin{multline*}
D(aP+bQ)=D\left(a\sum_{k}p_{k}X^{k}+b\sum_{k}q_{k}X^{k}\right)=D\left(\sum_{k}(ap_{k}+bq_{k})X^{k}\right)=\\
@@ -1928,16 +2021,20 @@ D(aP+bQ)=D\left(a\sum_{k}p_{k}X^{k}+b\sum_{k}q_{k}X^{k}\right)=D\left(\sum_{k}(a
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $(PQ)'=P'Q+PQ'$
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula
\begin{multline*}
D(PQ)=D\left(\left(\sum_{k}p_{k}X^{k}\right)\left(\sum_{k}q_{k}X^{k}\right)\right)=D\left(\sum_{k}\left(\sum_{i=0}^{k}p_{i}q_{k-i}\right)X^{k}\right)=\\
@@ -1951,16 +2048,20 @@ D(PQ)=D\left(\left(\sum_{k}p_{k}X^{k}\right)\left(\sum_{k}q_{k}X^{k}\right)\righ
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $(P^{n})'=nP^{n-1}P'$
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $P^{n-1}$
\end_inset
@@ -1993,7 +2094,11 @@ D(PQ)=D\left(\left(\sum_{k}p_{k}X^{k}\right)\left(\sum_{k}q_{k}X^{k}\right)\righ
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Dados un dominio
\begin_inset Formula $D$
@@ -2024,7 +2129,11 @@ Dados un dominio
\end_inset
.
-
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
\series bold
Demostración:
\series default
@@ -2137,6 +2246,11 @@ Demostración:
.
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Section
Divisibilidad en anillos de polinomios
\end_layout
@@ -2155,7 +2269,8 @@ Dado un anillo
\end_inset
es un cuerpo.
-\end_layout
+\begin_inset Note Comment
+status open
\begin_layout Description
\begin_inset Formula $1\implies2]$
@@ -2270,6 +2385,11 @@ Dado un anillo
son invertibles.
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Sean
\begin_inset Formula $D$
@@ -3150,22 +3270,6 @@ status open
\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
@@ -3208,7 +3312,12 @@ Si
\end_inset
.
- Esto está bien definido, pues si
+ Esto está bien definido
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+, pues si
\begin_inset Formula $b_{1}\sim b_{2}$
\end_inset
@@ -3220,28 +3329,17 @@ Si
\begin_inset Formula $(ab_{2})D^{*}=(aub_{1})D^{*}=\{ab_{1}uv\}_{v\in D^{*}}=\{ab_{1}v\}_{v\in D^{*}}=(ab_{1})D^{*}$
\end_inset
-.
- Además,
-\begin_inset Formula $a(b(cD^{*}))=(ab)(cD^{*})$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-\begin_layout Plain Layout
-
-
-\backslash
-end{samepage}
\end_layout
\end_inset
+.
+ Además,
+\begin_inset Formula $a(b(cD^{*}))=(ab)(cD^{*})$
+\end_inset
+.
\end_layout
\begin_layout Standard
@@ -3274,7 +3372,12 @@ Definimos
\end_inset
.
- Esto está bien definido, pues si
+ Esto está bien definido
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+, pues si
\begin_inset Formula $a_{1}p,a_{2}p\in D[X]$
\end_inset
@@ -3286,6 +3389,11 @@ Definimos
\begin_inset Formula $a_{1}^{-1}c(a_{1}p)=a_{2}^{-1}c(a_{2}p)$
\end_inset
+
+\end_layout
+
+\end_inset
+
.
Si
\begin_inset Formula $c(p)=aD^{*}$
@@ -3344,10 +3452,10 @@ Si
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $a\mid p$
\end_inset
@@ -3370,16 +3478,20 @@ Si
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $c(ap)=ac(p)$
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Si
\begin_inset Formula $a\in D$
\end_inset
@@ -3404,15 +3516,19 @@ Si
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $p\in D[X]\iff c(p)\in D$
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open
@@ -3429,6 +3545,7 @@ status open
Obvio.
\end_layout
+\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open
@@ -3572,6 +3689,11 @@ Sea
\end_layout
\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Un polinomio
\begin_inset Formula $p$
@@ -3810,10 +3932,10 @@ Si
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Sean
\begin_inset Formula $g,h\in K[X]$
\end_inset
@@ -3833,7 +3955,11 @@ Sean
es unidad.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Si
\begin_inset Formula $\text{gr}(f)>1$
@@ -3856,10 +3982,10 @@ Si
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Sean
\begin_inset Formula $a$
\end_inset
@@ -3887,7 +4013,11 @@ Sean
son unidades.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Si
\begin_inset Formula $\text{gr}(f)\in\{2,3\}$
@@ -3906,9 +4036,9 @@ Si
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open
@@ -3925,6 +4055,7 @@ status open
Es el contrarrecíproco de lo anterior.
\end_layout
+\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open
@@ -3983,6 +4114,11 @@ De haber
\end_layout
\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Si
\begin_inset Formula $D$
@@ -4021,7 +4157,11 @@ Si
\end_inset
.
- En efecto, sea
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+En efecto, sea
\begin_inset Formula $t=\frac{r}{s}$
\end_inset
@@ -4072,6 +4212,11 @@ Si
.
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
\series bold
@@ -4126,7 +4271,11 @@ Criterio de reducción:
\end_inset
.
-
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
\series bold
Demostración:
\series default
@@ -4202,6 +4351,11 @@ Demostración:
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
En particular, si
\begin_inset Formula $p\in\mathbb{Z}$
@@ -4276,7 +4430,11 @@ Criterio de Eisenstein:
\end_inset
.
-
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
\series bold
Demostración:
\series default
@@ -4353,6 +4511,11 @@ Demostración:
es análogo.
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Así:
\end_layout
@@ -4375,10 +4538,10 @@ Si
\end_inset
es irreducible.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $X^{n}-a$
\end_inset
@@ -4393,7 +4556,11 @@ Si
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Para
\begin_inset Formula $n\geq3$
@@ -4474,10 +4641,10 @@ de 1
\end_inset
es irreducible.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Usando el automorfismo de sustitución en
\begin_inset Formula $X+1$
\end_inset
@@ -4521,7 +4688,11 @@ Entonces
y podemos aplicar el criterio de Eisenstein.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Section
Polinomios en varias indeterminadas
\end_layout
@@ -4581,7 +4752,17 @@ polinomios en
\begin_inset Formula $n\in\mathbb{N}^{*}$
\end_inset
-, por inducción:
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+, por inducción
+\end_layout
+
+\end_inset
+
+:
\end_layout
\begin_layout Enumerate
@@ -4691,7 +4872,11 @@ con
\end_inset
.
-
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
\series bold
Demostración:
\series default
@@ -4772,17 +4957,6 @@ p=\sum_{i\in\mathbb{N}^{n}}p_{i}X_{1}^{i_{1}}\cdots X_{n}^{i_{n}}=\sum_{i\in\mat
.
\end_layout
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-begin{samepage}
-\end_layout
-
\end_inset
@@ -4897,22 +5071,6 @@ Dados un anillo conmutativo
\end_layout
\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-end{samepage}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
Así:
\end_layout
generated by cgit v1.2.3 (git 2.46.0) at 2025-10-27 08:21:40 +0000