Terminados apuntes de Álgebra Conmutativa

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+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\input{../defs}
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Section
+Cuerpos de fracciones
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{GyA}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $D\neq0$
+\end_inset
+
+ un dominio y 
+\begin_inset Formula $X\coloneqq D\times(D\setminus\{0\})$
+\end_inset
+
+, definimos la relación binaria 
+\begin_inset Formula 
+\[
+(a_{1},s_{1})\sim(a_{2},s_{2}):\iff a_{1}s_{2}=a_{2}s_{1}.
+\]
+
+\end_inset
+
+ Esta relación es de equivalencia.
+ Llamamos 
+\begin_inset Formula $a/s\coloneqq\frac{a}{s}\coloneqq[(a,s)]\in Q(D)\coloneqq X/\sim$
+\end_inset
+
+, y las operaciones
+\begin_inset Formula 
+\begin{align*}
+\frac{a_{1}}{s_{1}}+\frac{a_{2}}{s_{2}} & :=\frac{a_{1}s_{2}+a_{2}s_{1}}{s_{1}s_{2}}, & \frac{a_{1}}{s_{1}}\cdot\frac{a_{2}}{s_{2}} & :=\frac{a_{1}a_{2}}{s_{1}s_{2}},
+\end{align*}
+
+\end_inset
+
+están bien definidas.
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $a,b\in D$
+\end_inset
+
+ y 
+\begin_inset Formula $s,t\in D\setminus\{0\}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\frac{a}{s}=\frac{0}{1}\iff a=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\frac{a}{s}=\frac{1}{1}\iff a=s$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\frac{at}{st}=\frac{a}{s}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\frac{a}{s}=\frac{b}{s}\iff a=b$
+\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_layout Standard
+[...] 
+\begin_inset Formula $(Q(D),+,\cdot)$
+\end_inset
+
+ es un cuerpo llamado 
+\series bold
+cuerpo de fracciones
+\series default
+ o 
+\series bold
+de cocientes
+\series default
+ de 
+\begin_inset Formula $D$
+\end_inset
+
+ cuyo cero es 
+\begin_inset Formula $\frac{0}{1}$
+\end_inset
+
+ y cuyo uno es 
+\begin_inset Formula $\frac{1}{1}$
+\end_inset
+
+ .
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ es el cuerpo de fracciones de 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+.
+ [...] 
+\begin_inset Formula $u:D\to Q(D)$
+\end_inset
+
+ dada por 
+\begin_inset Formula $u(a)\coloneqq a/1$
+\end_inset
+
+ es un homomorfismo inyectivo, por lo que podemos ver a 
+\begin_inset Formula $D$
+\end_inset
+
+ como un subdominio de 
+\begin_inset Formula $Q(D)$
+\end_inset
+
+ identificando a cada 
+\begin_inset Formula $a\in D$
+\end_inset
+
+ con 
+\begin_inset Formula $a/1\in Q(D)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Propiedad universal del cuerpo de fracciones:
+\series default
+ Dados un dominio 
+\begin_inset Formula $D$
+\end_inset
+
+ y 
+\begin_inset Formula $u:D\to Q(D)$
+\end_inset
+
+ dada por 
+\begin_inset Formula $u(a)\coloneqq a/1$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $K$
+\end_inset
+
+ un cuerpo y 
+\begin_inset Formula $f:D\to K$
+\end_inset
+
+ un homomorfismo inyectivo, el único homomorfismo de cuerpos 
+\begin_inset Formula $\tilde{f}:Q(D)\to K$
+\end_inset
+
+ con 
+\begin_inset Formula $\tilde{f}\circ u=f$
+\end_inset
+
+ viene dado por 
+\begin_inset Formula $\tilde{f}(\frac{a}{s})=f(a)f(s)^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $K$
+\end_inset
+
+ un cuerpo no trivial y 
+\begin_inset Formula $g,h:Q(D)\to K$
+\end_inset
+
+ homomorfismos que coinciden en 
+\begin_inset Formula $D$
+\end_inset
+
+, entonces 
+\begin_inset Formula $g=h$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $F$
+\end_inset
+
+ un cuerpo no trivial y 
+\begin_inset Formula $v:D\to F$
+\end_inset
+
+ un homomorfismo inyectivo tal que para todo cuerpo 
+\begin_inset Formula $K$
+\end_inset
+
+ y homomorfismo inyectivo 
+\begin_inset Formula $f:D\to K$
+\end_inset
+
+ existe un único homomorfismo 
+\begin_inset Formula $\tilde{f}:F\to K$
+\end_inset
+
+ con 
+\begin_inset Formula $\tilde{f}\circ v=f$
+\end_inset
+
+, entonces existe un isomorfismo 
+\begin_inset Formula $\phi:F\to Q(D)$
+\end_inset
+
+ con 
+\begin_inset Formula $\phi\circ v=u$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $D$
+\end_inset
+
+ un dominio, 
+\begin_inset Formula $K$
+\end_inset
+
+ un cuerpo no trivial y 
+\begin_inset Formula $f:D\to K$
+\end_inset
+
+ un homomorfismo inyectivo, 
+\begin_inset Formula $K$
+\end_inset
+
+ contiene un subcuerpo isomorfo a 
+\begin_inset Formula $Q(D)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+De aquí, para 
+\begin_inset Formula $m\in\mathbb{Z}$
+\end_inset
+
+, 
+\begin_inset Formula $Q(\mathbb{Z}[\sqrt{m}])\cong\mathbb{Q}[\sqrt{m}]$
+\end_inset
+
+, lo que nos permite identificar los elementos de 
+\begin_inset Formula $Q(\mathbb{Z}[\sqrt{m}])$
+\end_inset
+
+ con los de 
+\begin_inset Formula $\mathbb{Q}[\sqrt{m}]$
+\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
+end{reminder}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Polinomios
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{GyA}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A$
+\end_inset
+
+ es un subanillo de 
+\begin_inset Formula $A[X]$
+\end_inset
+
+ identificando los elementos de 
+\begin_inset Formula $A$
+\end_inset
+
+ con los 
+\series bold
+polinomios constantes
+\series default
+, de la forma 
+\begin_inset Formula $P(X)=a_{0}$
+\end_inset
+
+.
+ Dado un ideal 
+\begin_inset Formula $I$
+\end_inset
+
+ de 
+\begin_inset Formula $A$
+\end_inset
+
+, 
+\begin_inset Formula $\{a_{0}+a_{1}X+\dots+a_{n}X^{n}\in A[X]\mid a_{0}\in I\}$
+\end_inset
+
+ e 
+\begin_inset Formula $I[X]\coloneqq\{a_{0}+a_{1}X+\dots+a_{n}X^{n}\in A[X]\mid a_{0},\dots,a_{n}\in I\}$
+\end_inset
+
+ son ideales de 
+\begin_inset Formula $A[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dado 
+\begin_inset Formula $p\coloneqq\sum_{k\in\mathbb{N}}p_{k}X^{k}\in A[X]\setminus\{0\}$
+\end_inset
+
+, llamamos 
+\series bold
+grado
+\series default
+ de 
+\begin_inset Formula $p$
+\end_inset
+
+ a 
+\begin_inset Formula $\text{gr}(p)\coloneqq\max\{k\in\mathbb{N}\mid p_{k}\neq0\}$
+\end_inset
+
+, 
+\series bold
+coeficiente
+\series default
+ de 
+\series bold
+grado
+\series default
+ 
+\begin_inset Formula $k$
+\end_inset
+
+ de 
+\begin_inset Formula $p$
+\end_inset
+
+ a 
+\begin_inset Formula $p_{k}$
+\end_inset
+
+, 
+\series bold
+coeficiente independiente
+\series default
+ al de grado 0 y 
+\series bold
+coeficiente principal
+\series default
+ al de grado 
+\begin_inset Formula $\text{gr}(p)$
+\end_inset
+
+.
+ Un polinomio es 
+\series bold
+mónico
+\series default
+ si su coeficiente principal es 1.
+ El polinomio 0 tiene grado 
+\begin_inset Formula $-\infty$
+\end_inset
+
+ por convención.
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+monomio
+\series default
+ es un polinomio de la forma 
+\begin_inset Formula $aX^{n}$
+\end_inset
+
+ con 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ y 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+.
+ Todo polinomio en 
+\begin_inset Formula $A[X]$
+\end_inset
+
+ se escribe como suma finita de monomios de distinto grado de forma única
+ salvo orden.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $P,Q\in A[X]\setminus\{0\}$
+\end_inset
+
+ tienen coeficientes principales respectivos 
+\begin_inset Formula $p$
+\end_inset
+
+ y 
+\begin_inset Formula $q$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{gr}(P+Q)\leq\max\{\text{gr}(P),\text{gr}(Q)\}$
+\end_inset
+
+, con desigualdad estricta si y sólo si 
+\begin_inset Formula $\text{gr}(P)=\text{gr}(Q)$
+\end_inset
+
+ y 
+\begin_inset Formula $p+q=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{gr}(PQ)\leq\text{gr}(P)+\text{gr}(Q)$
+\end_inset
+
+, con igualdad si y sólo si 
+\begin_inset Formula $pq\neq0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A[X]$
+\end_inset
+
+ no es un cuerpo.
+ Es un dominio si y sólo si lo es 
+\begin_inset Formula $A$
+\end_inset
+
+, en cuyo caso llamamos 
+\series bold
+cuerpo de las funciones racionales
+\series default
+ sobre 
+\begin_inset Formula $A$
+\end_inset
+
+ al cuerpo de fracciones de 
+\begin_inset Formula $A[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+[...] 
+\series bold
+Propiedad universal del anillo de polinomios
+\series default
+ (
+\series bold
+PUAP
+\series default
+)
+\series bold
+:
+\series default
+ Sean 
+\begin_inset Formula $A$
+\end_inset
+
+ un anillo y 
+\begin_inset Formula $u:A\to A[X]$
+\end_inset
+
+ el homomorfismo inclusión:
+\end_layout
+
+\begin_layout Enumerate
+Para cada homomorfismo de anillos conmutativos 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ y 
+\begin_inset Formula $b\in B$
+\end_inset
+
+, el único homomorfismo 
+\begin_inset Formula $\tilde{f}:A[X]\to B$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\tilde{f}(X)=b$
+\end_inset
+
+ y 
+\begin_inset Formula $\tilde{f}\circ u=f$
+\end_inset
+
+ es
+\begin_inset Formula 
+\[
+\tilde{f}\left(\sum_{n}p_{n}X^{n}\right):=\sum_{n}f(p_{n})b^{n}.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A[X]$
+\end_inset
+
+ y 
+\begin_inset Formula $u$
+\end_inset
+
+ están determinados salvo isomorfismos por la propiedad universal: dados
+ un homomorfismo de anillos 
+\begin_inset Formula $v:A\to P$
+\end_inset
+
+ y 
+\begin_inset Formula $t\in P$
+\end_inset
+
+ tales que, para cada homomorfismo de anillos 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ y 
+\begin_inset Formula $b\in B$
+\end_inset
+
+, existe un único 
+\begin_inset Formula $\tilde{f}:P\to B$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\tilde{f}\circ v=f$
+\end_inset
+
+ y 
+\begin_inset Formula $\tilde{f}(t)=b$
+\end_inset
+
+, existe un isomorfismo 
+\begin_inset Formula $\phi:A[X]\to P$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\phi\circ u=v$
+\end_inset
+
+ y 
+\begin_inset Formula $\phi(X)=t$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Así:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es un subanillo de 
+\begin_inset Formula $B$
+\end_inset
+
+ y 
+\begin_inset Formula $b\in B$
+\end_inset
+
+, el 
+\series bold
+homomorfismo de sustitución
+\series default
+ o 
+\series bold
+de evaluación
+\series default
+ en 
+\begin_inset Formula $b$
+\end_inset
+
+ es 
+\begin_inset Formula $S_{b}:A[X]\to B$
+\end_inset
+
+ dado por
+\begin_inset Formula 
+\[
+S_{b}(p):=p(b):=\sum_{n}p_{n}b^{n},
+\]
+
+\end_inset
+
+y su imagen es el subanillo generado por 
+\begin_inset Formula $A\cup\{b\}$
+\end_inset
+
+, llamado 
+\begin_inset Formula $A[b]$
+\end_inset
+
+.
+ Todo 
+\begin_inset Formula $p\in A[X]$
+\end_inset
+
+ induce una 
+\series bold
+función polinómica
+\series default
+ 
+\begin_inset Formula $\hat{p}:B\to B$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\hat{p}(b)\coloneqq S_{b}(p)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Dado 
+\begin_inset Formula $a\in A$
+\end_inset
+
+, el homomorfismo de sustitución 
+\begin_inset Formula $S_{X+a}$
+\end_inset
+
+ es un automorfismo de 
+\begin_inset Formula $A[X]$
+\end_inset
+
+ con inverso 
+\begin_inset Formula $S_{X-a}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es un anillo conmutativo, 
+\begin_inset Formula $\frac{A[X]}{(X)}\cong A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Todo homomorfismo de anillos 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ induce un homomorfismo 
+\begin_inset Formula $\hat{f}:A[X]\to B[X]$
+\end_inset
+
+ dado por
+\begin_inset Formula 
+\[
+\hat{f}(p)=\sum_{n}f(p_{n})X^{n},
+\]
+
+\end_inset
+
+que es inyectivo o suprayectivo si lo es 
+\begin_inset Formula $f$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es un subanillo de 
+\begin_inset Formula $B$
+\end_inset
+
+, 
+\begin_inset Formula $A[X]$
+\end_inset
+
+ lo es de 
+\begin_inset Formula $B[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $I$
+\end_inset
+
+ es un ideal de 
+\begin_inset Formula $A$
+\end_inset
+
+, el 
+\series bold
+homomorfismo de reducción de coeficientes módulo 
+\begin_inset Formula $I$
+\end_inset
+
+
+\series default
+ es 
+\begin_inset Formula $\tilde{\pi}:A[X]\to(A/I)[X]$
+\end_inset
+
+ dado por
+\begin_inset Formula 
+\[
+\tilde{\pi}(p):=\sum_{n}(p_{n}+I)X^{n}.
+\]
+
+\end_inset
+
+Su núcleo es 
+\begin_inset Formula $I[X]$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $(A/I)[X]\cong\frac{A[X]}{I[X]}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{reminder}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Newpage pagebreak
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Descomposiciones de polinomios en dominios
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{GyA}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $f,g\in A[X]$
+\end_inset
+
+, si el coeficiente principal de 
+\begin_inset Formula $g$
+\end_inset
+
+ es invertible en 
+\begin_inset Formula $A$
+\end_inset
+
+, existen dos únicos polinomios 
+\begin_inset Formula $q,r\in A[X]$
+\end_inset
+
+, llamados respectivamente 
+\series bold
+cociente
+\series default
+ y 
+\series bold
+resto
+\series default
+ de la 
+\series bold
+división
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ entre 
+\begin_inset Formula $g$
+\end_inset
+
+, tales que 
+\begin_inset Formula $f=gq+r$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{gr}(r)<\text{gr}(g)$
+\end_inset
+
+ [...].
+ En particular, el grado es una función euclídea.
+ 
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema del resto:
+\series default
+ Dados 
+\begin_inset Formula $f\in A[X]$
+\end_inset
+
+ y 
+\begin_inset Formula $a\in A$
+\end_inset
+
+, el resto de 
+\begin_inset Formula $f$
+\end_inset
+
+ entre 
+\begin_inset Formula $X-a$
+\end_inset
+
+ es 
+\begin_inset Formula $f(a)$
+\end_inset
+
+.
+ De aquí se obtiene el 
+\series bold
+teorema de Ruffini
+\series default
+, que dice que 
+\begin_inset Formula $f$
+\end_inset
+
+ es divisible por 
+\begin_inset Formula $X-a$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $f(a)=0$
+\end_inset
+
+, en cuyo caso 
+\begin_inset Formula $a$
+\end_inset
+
+ es una 
+\series bold
+raíz
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $f\in A[X]\setminus\{0\}$
+\end_inset
+
+ y 
+\begin_inset Formula $a\in A$
+\end_inset
+
+, existe 
+\begin_inset Formula $m\coloneqq\max\{k\in\mathbb{N}\mid(X-a)^{k}\mid f\}$
+\end_inset
+
+.
+ Llamamos a 
+\begin_inset Formula $m$
+\end_inset
+
+ 
+\series bold
+multiplicidad
+\series default
+ de 
+\begin_inset Formula $a$
+\end_inset
+
+ en 
+\begin_inset Formula $f$
+\end_inset
+
+, y 
+\begin_inset Formula $a$
+\end_inset
+
+ es raíz de 
+\begin_inset Formula $f$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $m\geq1$
+\end_inset
+
+.
+ Decimos que 
+\begin_inset Formula $a$
+\end_inset
+
+ es una 
+\series bold
+raíz simple
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ si 
+\begin_inset Formula $m=1$
+\end_inset
+
+ y que es una 
+\series bold
+raíz compuesta
+\series default
+ si 
+\begin_inset Formula $m>1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+La multiplicidad de 
+\begin_inset Formula $a$
+\end_inset
+
+ en 
+\begin_inset Formula $f$
+\end_inset
+
+ es el único natural 
+\begin_inset Formula $m$
+\end_inset
+
+ tal que 
+\begin_inset Formula $f=(X-a)^{m}g$
+\end_inset
+
+ para algún 
+\begin_inset Formula $g\in A[X]$
+\end_inset
+
+ del que 
+\begin_inset Formula $a$
+\end_inset
+
+ no es raíz.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $D$
+\end_inset
+
+ es un dominio, 
+\begin_inset Formula $f\in D[X]\setminus\{0\}$
+\end_inset
+
+, 
+\begin_inset Formula $a_{1},\dots,a_{n}$
+\end_inset
+
+ son 
+\begin_inset Formula $n$
+\end_inset
+
+ elementos de 
+\begin_inset Formula $D$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha_{1},\dots,\alpha_{n}\in\mathbb{Z}^{>0}$
+\end_inset
+
+ con 
+\begin_inset Formula $(X-a_{k})^{\alpha_{k}}\mid f$
+\end_inset
+
+ para cada 
+\begin_inset Formula $k$
+\end_inset
+
+, entonces 
+\begin_inset Formula $(X-a_{1})^{\alpha_{1}}\cdots(X-a_{n})^{\alpha_{n}}\mid f$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $\sum_{k=1}^{n}\alpha_{k}\leq\text{gr}(f)$
+\end_inset
+
+ y, en particular, la suma de las multiplicidades de las raíces de 
+\begin_inset Formula $f$
+\end_inset
+
+, y el número de raíces, no son superiores a 
+\begin_inset Formula $\text{gr}(f)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Principio de las identidades polinómicas:
+\series default
+ Sea 
+\begin_inset Formula $D$
+\end_inset
+
+ un dominio:
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $f,g\in D[X]$
+\end_inset
+
+, si las funciones polinómicas 
+\begin_inset Formula $f,g:D\to D$
+\end_inset
+
+ coinciden en 
+\begin_inset Formula $m$
+\end_inset
+
+ elementos de 
+\begin_inset Formula $D$
+\end_inset
+
+ con 
+\begin_inset Formula $m>\text{gr}(f),\text{gr}(g)$
+\end_inset
+
+, los polinomios 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $g$
+\end_inset
+
+ son iguales.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $D$
+\end_inset
+
+ es infinito si y sólo si cualquier par de polinomios distintos en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+ define dos funciones polinómicas distintas en 
+\begin_inset Formula $D$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Como ejemplo de lo anterior, por el teorema pequeño de Fermat, dado un primo
+ 
+\begin_inset Formula $p$
+\end_inset
+
+, todos los elementos de 
+\begin_inset Formula $\mathbb{Z}_{p}$
+\end_inset
+
+ son raíces de 0 y 
+\begin_inset Formula $X^{p}-X$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dado un anillo conmutativo 
+\begin_inset Formula $A$
+\end_inset
+
+, definimos la 
+\series bold
+derivada
+\series default
+ de 
+\begin_inset Formula $P\coloneqq\sum_{k}a_{k}X^{k}\in A[X]$
+\end_inset
+
+ como 
+\begin_inset Formula $P'\coloneqq D(P)\coloneqq\sum_{k\geq1}ka_{k}X^{k-1}$
+\end_inset
+
+, y escribimos 
+\begin_inset Formula $P^{(0)}\coloneqq P$
+\end_inset
+
+ y 
+\begin_inset Formula $P^{(n+1)}\coloneqq P^{(n)\prime}$
+\end_inset
+
+.
+ Dados 
+\begin_inset Formula $a,b\in A$
+\end_inset
+
+ y 
+\begin_inset Formula $P,Q\in A[X]$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(aP+bQ)'=aP'+bQ'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(PQ)'=P'Q+PQ'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(P^{n})'=nP^{n-1}P'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dados un dominio 
+\begin_inset Formula $D$
+\end_inset
+
+ de característica 0, 
+\begin_inset Formula $P\in D[X]\setminus\{0\}$
+\end_inset
+
+ y 
+\begin_inset Formula $a\in D$
+\end_inset
+
+, la multiplicidad de 
+\begin_inset Formula $a$
+\end_inset
+
+ en 
+\begin_inset Formula $P$
+\end_inset
+
+ es el menor 
+\begin_inset Formula $m\in\mathbb{N}_{0}$
+\end_inset
+
+ con 
+\begin_inset Formula $P^{(m)}(a)\neq0$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Dado un anillo 
+\begin_inset Formula $A$
+\end_inset
+
+, 
+\begin_inset Formula $A[X]$
+\end_inset
+
+ es un dominio euclídeo si y sólo si es un DIP, si y sólo si 
+\begin_inset Formula $A$
+\end_inset
+
+ es un cuerpo.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $D$
+\end_inset
+
+ un dominio y 
+\begin_inset Formula $p\in D$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $p$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $D$
+\end_inset
+
+ si y sólo si lo es en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $p$
+\end_inset
+
+ es primo en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+, lo es en 
+\begin_inset Formula $D$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $D$
+\end_inset
+
+ es un DFU, 
+\begin_inset Formula $p$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $D$
+\end_inset
+
+ si y sólo si lo es en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+, si y sólo si es primo en 
+\begin_inset Formula $D$
+\end_inset
+
+, si y sólo si lo es en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $D$
+\end_inset
+
+ un DFU, definimos 
+\begin_inset Formula $\varphi:D\setminus0\to\mathbb{N}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\varphi(a)$
+\end_inset
+
+ es el número de factores irreducibles en la factorización por irreducibles
+ de 
+\begin_inset Formula $a$
+\end_inset
+
+ en 
+\begin_inset Formula $D$
+\end_inset
+
+, contando repetidos, y para 
+\begin_inset Formula $a,b\in D\setminus\{0\}$
+\end_inset
+
+, 
+\begin_inset Formula $\varphi(ab)=\varphi(a)+\varphi(b)$
+\end_inset
+
+ y 
+\begin_inset Formula $\varphi(a)=0\iff a\in D^{*}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $D$
+\end_inset
+
+ es un DFU, 
+\begin_inset Formula $K$
+\end_inset
+
+ es su cuerpo de fracciones y 
+\begin_inset Formula $f\in D[X]$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+, es irreducible en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+.
+ [...] 
+\begin_inset Formula $D$
+\end_inset
+
+ es un DFU si y sólo si lo es 
+\begin_inset Formula $D[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+[...] Si 
+\begin_inset Formula $D$
+\end_inset
+
+ es un DFU y 
+\begin_inset Formula $K$
+\end_inset
+
+ es su cuerpo de fracciones, definimos la relación de equivalencia en 
+\begin_inset Formula $K$
+\end_inset
+
+ 
+\begin_inset Formula $x\sim y:\iff\exists u\in D^{*}:y=ux$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $[x]=xD^{*}$
+\end_inset
+
+ y, en particular, si 
+\begin_inset Formula $x\in D$
+\end_inset
+
+, 
+\begin_inset Formula $[x]$
+\end_inset
+
+ es el conjunto de los asociados de 
+\begin_inset Formula $x$
+\end_inset
+
+ en 
+\begin_inset Formula $D$
+\end_inset
+
+.
+ Definimos 
+\begin_inset Formula $\cdot:K\times(K/\sim)\to K/\sim$
+\end_inset
+
+ como 
+\begin_inset Formula $a(bD^{*})=(ab)D^{*}$
+\end_inset
+
+.
+ Esto está bien definido.
+ Además, 
+\begin_inset Formula $a(b(cD^{*}))=(ab)(cD^{*})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Definimos 
+\begin_inset Formula $c:K[X]\to K/\sim$
+\end_inset
+
+ tal que, para 
+\begin_inset Formula $p\coloneqq\sum_{k\geq0}p_{k}X^{k}\in D[X]$
+\end_inset
+
+, 
+\begin_inset Formula $c(p)\coloneqq\{x\mid x=\text{mcd}_{k\geq0}p_{k}\}$
+\end_inset
+
+, y para 
+\begin_inset Formula $p\in K[X]$
+\end_inset
+
+, si 
+\begin_inset Formula $a\in D\setminus\{0\}$
+\end_inset
+
+ cumple 
+\begin_inset Formula $ap\in D[X]$
+\end_inset
+
+, 
+\begin_inset Formula $c(p)\coloneqq a^{-1}c(ap)$
+\end_inset
+
+.
+ Esto está bien definido.
+ Si 
+\begin_inset Formula $c(p)=aD^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $a$
+\end_inset
+
+ es el 
+\series bold
+contenido
+\series default
+ de 
+\begin_inset Formula $p$
+\end_inset
+
+ (
+\begin_inset Formula $a=c(p)$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $a\in K$
+\end_inset
+
+ y 
+\begin_inset Formula $p\in K[X]$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $a\in D$
+\end_inset
+
+ y 
+\begin_inset Formula $p\in D[X]$
+\end_inset
+
+, 
+\begin_inset Formula $a\mid p$
+\end_inset
+
+ en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $a\mid c(p)$
+\end_inset
+
+ en 
+\begin_inset Formula $D$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $c(ap)=ac(p)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $p\in D[X]\iff c(p)\in D$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Un polinomio 
+\begin_inset Formula $p$
+\end_inset
+
+ es 
+\series bold
+primitivo
+\series default
+ si 
+\begin_inset Formula $c(p)=1$
+\end_inset
+
+, esto es, si 
+\begin_inset Formula $p\in D[X]$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{mcd}_{k}p_{k}=1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Lema de Gauss:
+\series default
+ Para 
+\begin_inset Formula $f,g\in D[X]$
+\end_inset
+
+, 
+\begin_inset Formula $c(fg)=c(f)c(g)$
+\end_inset
+
+, y en particular 
+\begin_inset Formula $fg$
+\end_inset
+
+ es primitivo si y sólo si 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $g$
+\end_inset
+
+ lo son.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Dado 
+\begin_inset Formula $f\in D[X]\setminus D$
+\end_inset
+
+ primitivo, 
+\begin_inset Formula $f$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+ si y sólo si lo es en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\forall G,H\in K[X],(f=GH\implies\text{gr}(G)=0\lor\text{gr}(H)=0)$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\forall g,h\in D[X],(f=gh\implies\text{gr}(g)=0\lor\text{gr}(h)=0)$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Standard
+De aquí que si 
+\begin_inset Formula $D$
+\end_inset
+
+ es un DFU con cuerpo de fracciones 
+\begin_inset Formula $K$
+\end_inset
+
+, los irreducibles de 
+\begin_inset Formula $D[X]$
+\end_inset
+
+ son precisamente los de 
+\begin_inset Formula $D$
+\end_inset
+
+ y los polinomios primitivos de 
+\begin_inset Formula $D[X]\setminus D$
+\end_inset
+
+ irreducibles en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+[...] Sean 
+\begin_inset Formula $K$
+\end_inset
+
+ un cuerpo y 
+\begin_inset Formula $f\in K[X]$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\text{gr}(f)=1$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\text{gr}(f)>1$
+\end_inset
+
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ tiene una raíz en 
+\begin_inset Formula $K$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ no es irreducible en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\text{gr}(f)\in\{2,3\}$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+ si y sólo si no tiene raíces en 
+\begin_inset Formula $K$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $D$
+\end_inset
+
+ es un DFU con cuerpo de fracciones 
+\begin_inset Formula $K$
+\end_inset
+
+, 
+\begin_inset Formula $f\coloneqq\sum_{k}a_{k}X^{k}\in D[X]$
+\end_inset
+
+ y 
+\begin_inset Formula $n\coloneqq\text{gr}(f)$
+\end_inset
+
+, todas las raíces de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $K$
+\end_inset
+
+ son de la forma 
+\begin_inset Formula $\frac{r}{s}$
+\end_inset
+
+ con 
+\begin_inset Formula $r\mid a_{0}$
+\end_inset
+
+ y 
+\begin_inset Formula $s\mid a_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Criterio de reducción:
+\series default
+ Sean 
+\begin_inset Formula $\phi:D\to K$
+\end_inset
+
+ un homomorfismo de anillos donde 
+\begin_inset Formula $D$
+\end_inset
+
+ es un DFU y 
+\begin_inset Formula $K$
+\end_inset
+
+ es un cuerpo, 
+\begin_inset Formula $\hat{\phi}:D[X]\to K[X]$
+\end_inset
+
+ el homomorfismo inducido por 
+\begin_inset Formula $\phi$
+\end_inset
+
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ un polinomio primitivo de 
+\begin_inset Formula $D[X]\setminus D$
+\end_inset
+
+, si 
+\begin_inset Formula $\hat{\phi}(f)$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{gr}(\hat{\phi}(f))=\text{gr}(f)$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+En particular, si 
+\begin_inset Formula $p\in\mathbb{Z}$
+\end_inset
+
+ es primo, 
+\begin_inset Formula $f\coloneqq\sum_{k}a_{k}X^{k}\in\mathbb{Z}[X]$
+\end_inset
+
+ es primitivo, 
+\begin_inset Formula $n\coloneqq\text{gr}(f)$
+\end_inset
+
+, 
+\begin_inset Formula $p\nmid a_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $\mathbb{Z}_{p}[X]$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $\mathbb{Z}[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Criterio de Eisenstein:
+\series default
+ Sean 
+\begin_inset Formula $D$
+\end_inset
+
+ un DFU, 
+\begin_inset Formula $f\coloneqq\sum_{k}a_{k}X^{k}\in D[X]$
+\end_inset
+
+ primitivo y 
+\begin_inset Formula $n\coloneqq\text{gr}f$
+\end_inset
+
+, si existe un irreducible 
+\begin_inset Formula $p\in D$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall k\in\{0,\dots,n-1\},p\mid a_{k}$
+\end_inset
+
+ y 
+\begin_inset Formula $p^{2}\nmid a_{0}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Así:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $a\in\mathbb{Z}$
+\end_inset
+
+ y existe 
+\begin_inset Formula $p\in\mathbb{Z}$
+\end_inset
+
+ cuya multiplicidad en 
+\begin_inset Formula $a$
+\end_inset
+
+ es 1, 
+\begin_inset Formula $X^{n}-a$
+\end_inset
+
+ es irreducible.
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $n\geq3$
+\end_inset
+
+, llamamos 
+\series bold
+raíces 
+\begin_inset Formula $n$
+\end_inset
+
+-ésimas de la unidad
+\series default
+ o 
+\series bold
+de 1
+\series default
+ a las raíces de 
+\begin_inset Formula $X^{n}-1$
+\end_inset
+
+ en 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+, que son los 
+\begin_inset Formula $n$
+\end_inset
+
+ vértices del 
+\begin_inset Formula $n$
+\end_inset
+
+-ágono regular inscrito en el círculo unidad de 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+ con un vértice en el 1.
+ 
+\begin_inset Formula $X^{n}-1=(X-1)\Phi_{n}(X)$
+\end_inset
+
+, donde 
+\begin_inset Formula $\Phi_{n}(X)\coloneqq X^{n-1}+X^{n-2}+\dots+X+1$
+\end_inset
+
+ es el 
+\series bold
+
+\begin_inset Formula $n$
+\end_inset
+
+-ésimo polinomio ciclotómico
+\series default
+ y sus raíces en 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+ son las raíces 
+\begin_inset Formula $n$
+\end_inset
+
+-ésimas de 1 distintas de 1.
+ En 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+, 
+\begin_inset Formula $X+1\mid\Phi_{4}(X)$
+\end_inset
+
+, pero si 
+\begin_inset Formula $n$
+\end_inset
+
+ es primo, 
+\begin_inset Formula $\Phi_{n}(X)$
+\end_inset
+
+ es irreducible.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{reminder}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Polinomios en varias indeterminadas
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{GyA}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dados un anillo conmutativo 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $n\geq2$
+\end_inset
+
+, definimos el 
+\series bold
+anillo de polinomios
+\series default
+ en 
+\begin_inset Formula $n$
+\end_inset
+
+ indeterminadas con coeficientes en 
+\begin_inset Formula $A$
+\end_inset
+
+ como 
+\begin_inset Formula $A[X_{1},\dots,X_{n}]\coloneqq A[X_{1},\dots,X_{n-1}][X_{n}]$
+\end_inset
+
+.
+ Llamamos 
+\series bold
+indeterminadas
+\series default
+ a los símbolos 
+\begin_inset Formula $X_{1},\dots,X_{n}$
+\end_inset
+
+ y 
+\series bold
+polinomios en 
+\begin_inset Formula $n$
+\end_inset
+
+ indeterminadas
+\series default
+ a los elementos de 
+\begin_inset Formula $A[X_{1},\dots,X_{n}]$
+\end_inset
+
+.
+ Dados un anillo conmutativo 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $n\in\mathbb{N}^{*}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ no es un cuerpo.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ es un dominio si y sólo si lo es 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es un dominio, 
+\begin_inset Formula $A[X_{1},\dots,X_{n}]^{*}=A^{*}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ es un DFU si y sólo si lo es 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ es un DIP si y sólo si 
+\begin_inset Formula $n=1$
+\end_inset
+
+ y 
+\begin_inset Formula $A$
+\end_inset
+
+ es un cuerpo.
+\end_layout
+
+\begin_layout Standard
+Dados 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ e 
+\begin_inset Formula $i\coloneqq(i_{1},\dots,i_{n})\in\mathbb{N}^{n}$
+\end_inset
+
+, llamamos a 
+\begin_inset Formula $aX_{1}^{i_{1}}\cdots X_{n}^{i_{n}}\in A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ 
+\series bold
+monomio
+\series default
+ de 
+\series bold
+tipo
+\series default
+ 
+\begin_inset Formula $i$
+\end_inset
+
+ y coeficiente 
+\begin_inset Formula $a$
+\end_inset
+
+.
+ Todo 
+\begin_inset Formula $p\in A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ se escribe de forma única como suma de monomios de distinto tipo,
+\begin_inset Formula 
+\[
+p:=\sum_{i\in\mathbb{N}^{n}}p_{i}X_{1}^{i_{1}}\cdots X_{n}^{i_{n}},
+\]
+
+\end_inset
+
+con 
+\begin_inset Formula $p_{i}=0$
+\end_inset
+
+ para casi todo 
+\begin_inset Formula $i\in\mathbb{N}^{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+PUAP en 
+\begin_inset Formula $n$
+\end_inset
+
+ indeterminadas:
+\series default
+ Sean 
+\begin_inset Formula $A$
+\end_inset
+
+ un anillo conmutativo, 
+\begin_inset Formula $n\in\mathbb{N}^{*}$
+\end_inset
+
+ y 
+\begin_inset Formula $u:A\to A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ la inclusión:
+\end_layout
+
+\begin_layout Enumerate
+Dados un homomorfismo de anillos 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ y 
+\begin_inset Formula $b_{1},\dots,b_{n}\in B$
+\end_inset
+
+, existe un único homomorfismo de anillos 
+\begin_inset Formula $\tilde{f}:A[X_{1},\dots,X_{n}]\to B$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\tilde{f}\circ u=f$
+\end_inset
+
+ y 
+\begin_inset Formula $\tilde{f}(X_{k})=b_{k}$
+\end_inset
+
+ para 
+\begin_inset Formula $k\in\{1,\dots,n\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Dados un anillo conmutativo 
+\begin_inset Formula $P$
+\end_inset
+
+, 
+\begin_inset Formula $T_{1},\dots,T_{n}\in P$
+\end_inset
+
+ y un homomorfismo 
+\begin_inset Formula $v:A\to P$
+\end_inset
+
+ tales que, dados un homomorfismo de anillos 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ y 
+\begin_inset Formula $b_{1},\dots,b_{n}\in B$
+\end_inset
+
+, existe un único homomorfismo 
+\begin_inset Formula $\tilde{f}:P\to B$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\tilde{f}\circ v=f$
+\end_inset
+
+ y 
+\begin_inset Formula $\tilde{f}(T_{k})=b_{k}$
+\end_inset
+
+ para 
+\begin_inset Formula $k\in\{1,\dots,n\}$
+\end_inset
+
+, existe un isomorfismo 
+\begin_inset Formula $\phi:A[X_{1},\dots,X_{n}]\to P$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\phi\circ u=v$
+\end_inset
+
+ y 
+\begin_inset Formula $\phi(X_{k})=T_{k}$
+\end_inset
+
+ para cada 
+\begin_inset Formula $k\in\{1,\dots,n\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Así:
+\end_layout
+
+\begin_layout Enumerate
+Dados dos anillos conmutativos 
+\begin_inset Formula $A\subseteq B$
+\end_inset
+
+ y 
+\begin_inset Formula $b_{1},\dots,b_{n}\in B$
+\end_inset
+
+, el 
+\series bold
+homomorfismo de sustitución
+\series default
+ 
+\begin_inset Formula $S:A[X_{1},\dots,X_{n}]\to B$
+\end_inset
+
+ viene dado por 
+\begin_inset Formula $p(b_{1},\dots,b_{n})\coloneqq S(p)\coloneqq\sum_{i\in\mathbb{N}^{n}}p_{i}b_{1}^{i_{1}}\cdots b_{n}^{i_{n}}$
+\end_inset
+
+.
+ Su imagen es el subanillo de 
+\begin_inset Formula $B$
+\end_inset
+
+ generado por 
+\begin_inset Formula $A\cup\{b_{1},\dots,b_{n}\}$
+\end_inset
+
+, 
+\begin_inset Formula $A[b_{1},\dots,b_{n}]$
+\end_inset
+
+, y dados dos homomorfismos de anillos 
+\begin_inset Formula $f,g:A[b_{1},\dots,b_{n}]\to C$
+\end_inset
+
+, 
+\begin_inset Formula $f=g$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $f|_{A}=g|_{A}$
+\end_inset
+
+ y 
+\begin_inset Formula $f(b_{k})=g(b_{k})$
+\end_inset
+
+ para todo 
+\begin_inset Formula $k$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $A$
+\end_inset
+
+ un anillo y 
+\begin_inset Formula $\sigma$
+\end_inset
+
+ una permutación de 
+\begin_inset Formula $\mathbb{N}_{n}$
+\end_inset
+
+ con inversa 
+\begin_inset Formula $\tau\coloneqq\sigma^{-1}$
+\end_inset
+
+, tomando 
+\begin_inset Formula $B=A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ y 
+\begin_inset Formula $b_{k}=X_{\sigma(k)}$
+\end_inset
+
+ en el punto anterior obtenemos un automorfismo 
+\begin_inset Formula $\hat{\sigma}$
+\end_inset
+
+ en 
+\begin_inset Formula $A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ con inversa 
+\begin_inset Formula $\hat{\tau}$
+\end_inset
+
+ que permuta las indeterminadas.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A[X_{1},\dots,X_{n},Y_{1},\dots,Y_{m}]\cong A[X_{1},\dots,X_{n}][Y_{1},\dots,Y_{m}]\cong A[Y_{1},\dots,Y_{m}][X_{1},\dots,X_{n}]$
+\end_inset
+
+, por lo que en la práctica no distinguimos entre estos anillos.
+\end_layout
+
+\begin_layout Enumerate
+Todo homomorfismo de anillos conmutativos 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ induce un homomorfismo 
+\begin_inset Formula $\hat{f}:A[X_{1},\dots,X_{n}]\to B[X_{1},\dots,X_{n}]$
+\end_inset
+
+ dado por 
+\begin_inset Formula $\hat{f}(p)\coloneqq\sum_{i\in\mathbb{N}^{n}}f(p_{i})X_{1}^{i_{1}}\cdots X_{n}^{i_{n}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+grado
+\series default
+ de un monomio 
+\begin_inset Formula $aX_{1}^{i_{1}}\cdots X_{n}^{i_{n}}$
+\end_inset
+
+ a 
+\begin_inset Formula $i_{1}+\dots+i_{n}$
+\end_inset
+
+, y grado de 
+\begin_inset Formula $p\in A[X_{1},\dots,X_{n}]\setminus0$
+\end_inset
+
+, 
+\begin_inset Formula $\text{gr}(p)$
+\end_inset
+
+, al mayor de los grados de los monomios no nulos en la expresión por monomios
+ de 
+\begin_inset Formula $p$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\text{gr}(p+q)\leq\max\{\text{gr}(p),\text{gr}(q)\}$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{gr}(pq)\leq\text{gr}(p)+\text{gr}(q)$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Standard
+Un polinomio es 
+\series bold
+homogéneo
+\series default
+ de grado 
+\begin_inset Formula $n$
+\end_inset
+
+ si es suma de monomios de grado 
+\begin_inset Formula $n$
+\end_inset
+
+.
+ Todo polinomio se escribe de modo único como suma de polinomios homogéneos
+ de distintos grados, sin más que agrupar los monomios de igual grado en
+ la expresión como suma de monomios.
+ Así, si 
+\begin_inset Formula $D$
+\end_inset
+
+ es un dominio, 
+\begin_inset Formula $\text{gr}(pq)=\text{gr}(p)+\text{gr}(q)$
+\end_inset
+
+ para cualesquiera 
+\begin_inset Formula $p,q\in D[X_{1},\dots,X_{n}]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{reminder}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document