diff options
| -rw-r--r-- | ealg/n1.lyx | 263 |
1 files changed, 203 insertions, 60 deletions
diff --git a/ealg/n1.lyx b/ealg/n1.lyx index 54cf78e..b05e31c 100644 --- a/ealg/n1.lyx +++ b/ealg/n1.lyx @@ -1739,6 +1739,47 @@ Divisibilidad \end_layout \begin_layout Standard +Si +\begin_inset Formula $p\in A[X]$ +\end_inset + + está formado por subsecuencias proporcionales, es decir, si viene dado + por +\begin_inset Formula +\[ +(0,\dots,0,\alpha_{1}a_{0},\dots,\alpha_{1}a_{k},0,\dots,0,\alpha_{2}a_{0},\dots,\alpha_{2}a_{k},\dots,0,\dots,0,\alpha_{t}a_{0},\dots,\alpha_{t}a_{k},0,\dots), +\] + +\end_inset + +donde +\begin_inset Formula $\alpha_{1},\dots,\alpha_{t},a_{0},\dots a_{k}\in A$ +\end_inset + + con +\begin_inset Formula $a_{0},a_{k}\neq0$ +\end_inset + +, sea +\begin_inset Formula $n_{i}$ +\end_inset + + la posición de +\begin_inset Formula $\alpha_{i}$ +\end_inset + + para cada +\begin_inset Formula $i$ +\end_inset + +, entonces +\begin_inset Formula $p=(a_{0}+a_{1}X+\dots+a_{k}X^{k})(\alpha_{1}X^{n_{1}}+\dots+\alpha_{t}X^{n_{t}})$ +\end_inset + +. +\end_layout + +\begin_layout Standard \begin_inset ERT status open @@ -1845,6 +1886,23 @@ primitivo \end_inset . + [...] 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 + + [...]. \end_layout \begin_layout Standard @@ -2131,6 +2189,85 @@ que es irreducible por Eisenstein porque . \end_layout +\begin_layout Standard +Un polinomio +\begin_inset Formula $p\in K[X]$ +\end_inset + + de grado +\begin_inset Formula $n$ +\end_inset + + es +\series bold +recíproco +\series default + si para +\begin_inset Formula $i\in\{0,\dots,n\}$ +\end_inset + + es +\begin_inset Formula $p_{i}=p_{n-i}$ +\end_inset + +. + Si +\begin_inset Formula $K$ +\end_inset + + es un cuerpo y +\begin_inset Formula $n$ +\end_inset + + es par, las raíces no nulas de +\begin_inset Formula $p(X)$ +\end_inset + + son los ceros de +\begin_inset Formula $f(x):=p(x)/x^{n/2}:K^{*}\to K$ +\end_inset + +, que será de la forma +\begin_inset Formula +\[ +f(x)=p_{0}x^{k}+\dots+p_{k-1}x+p_{k}+p_{k-1}x^{-1}+\dots+p_{0}x^{-k}, +\] + +\end_inset + +donde +\begin_inset Formula $k:=n/2$ +\end_inset + +. + Haciendo el cambio de variable +\begin_inset Formula $y:=x+x^{-1}$ +\end_inset + + nos queda una función polinómica de grado +\begin_inset Formula $k$ +\end_inset + + y hemos reducido el grado a la mitad. + Para hacer el cambio, calculamos +\begin_inset Formula $y^{2},y^{3},\dots,y^{k}$ +\end_inset + +, sustituimos +\begin_inset Formula $p_{0}(x^{k}+x^{-k})$ +\end_inset + + por +\begin_inset Formula $p_{0}y^{k}$ +\end_inset + + más un polinomio de grado menor, hacemos lo propio con el grado +\begin_inset Formula $k-1$ +\end_inset + +, etc. +\end_layout + \begin_layout Section Factorización en \begin_inset Formula $\mathbb{C}[X]$ @@ -2388,6 +2525,72 @@ Demostración: La segunda parte se obtiene por inducción. \end_layout +\begin_layout Standard +Las raíces de +\begin_inset Formula $X^{n}-z\in\mathbb{C}[X]$ +\end_inset + + son las raíces +\begin_inset Formula $n$ +\end_inset + +-ésimas de +\begin_inset Formula $z$ +\end_inset + +. + Si +\begin_inset Formula $z=re^{i\theta}$ +\end_inset + + con +\begin_inset Formula $r,\theta\in\mathbb{R}$ +\end_inset + +, estas son de la forma +\begin_inset Formula $\sqrt[n]{r}\omega^{k}$ +\end_inset + +, con +\begin_inset Formula $\omega=e^{2\pi i/n}$ +\end_inset + +. + Para +\begin_inset Formula $n\in\mathbb{N}^{*}$ +\end_inset + +, una +\series bold +raíz +\begin_inset Formula $n$ +\end_inset + +-ésima de la unidad +\series default + es una de +\begin_inset Formula $X^{n}-1$ +\end_inset + + en +\begin_inset Formula $\mathbb{C}$ +\end_inset + +, y es +\series bold +primitiva +\series default + si no es raíz +\begin_inset Formula $k$ +\end_inset + +-ésima de la unidad para un +\begin_inset Formula $k<n$ +\end_inset + +. +\end_layout + \begin_layout Section Polinomios en varias variables \end_layout @@ -2538,34 +2741,6 @@ Dados un homomorfismo de anillos \end_layout \begin_layout Standard -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -eremember -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -sremember{GyA} -\end_layout - -\end_inset - Así: \end_layout @@ -2971,38 +3146,6 @@ grado \end_layout \begin_layout Standard -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -eremember -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -sremember{GyA} -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard Un polinomio es \series bold homogéneo |