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-rw-r--r--ealg/n1.lyx78
1 files changed, 39 insertions, 39 deletions
diff --git a/ealg/n1.lyx b/ealg/n1.lyx
index a5d022d..c0fcd21 100644
--- a/ealg/n1.lyx
+++ b/ealg/n1.lyx
@@ -211,7 +211,7 @@ polinomios constantes
\begin_layout Standard
Dado
-\begin_inset Formula $p:=\sum_{k\in\mathbb{N}}p_{k}X^{k}\in A[X]\setminus\{0\}$
+\begin_inset Formula $p\coloneqq \sum_{k\in\mathbb{N}}p_{k}X^{k}\in A[X]\setminus\{0\}$
\end_inset
, llamamos
@@ -223,7 +223,7 @@ grado
\end_inset
a
-\begin_inset Formula $\text{gr}(p):=\max\{k\in\mathbb{N}\mid p_{k}\neq0\}$
+\begin_inset Formula $\text{gr}(p)\coloneqq \max\{k\in\mathbb{N}\mid p_{k}\neq0\}$
\end_inset
,
@@ -831,7 +831,7 @@ euclídea
\end_layout
\begin_layout Enumerate
-\begin_inset Formula $\forall a\in D,b\in D\setminus\{0\},\exists q,r\in D\mid (a=bq+r\land(r=0\lor\delta(r)<\delta(b)))$
+\begin_inset Formula $\forall a\in D,b\in D\setminus\{0\},\exists q,r\in D:(a=bq+r\land(r=0\lor\delta(r)<\delta(b)))$
\end_inset
.
@@ -968,7 +968,7 @@ Para
\end_inset
, existe
-\begin_inset Formula $m:=\max\{k\in\mathbb{N}\mid (X-a)^{k}\mid f\}$
+\begin_inset Formula $m\coloneqq \max\{k\in\mathbb{N}\mid (X-a)^{k}\mid f\}$
\end_inset
.
@@ -1155,19 +1155,19 @@ Dado un anillo [...]
derivada
\series default
de
-\begin_inset Formula $P:=\sum_{k}a_{k}X^{k}\in A[X]$
+\begin_inset Formula $P\coloneqq \sum_{k}a_{k}X^{k}\in A[X]$
\end_inset
como
-\begin_inset Formula $P':=[...]:=\sum_{k\geq1}ka_{k}X^{k-1}$
+\begin_inset Formula $P'\coloneqq [...]\coloneqq \sum_{k\geq1}ka_{k}X^{k-1}$
\end_inset
, y escribimos
-\begin_inset Formula $P^{(0)}:=P$
+\begin_inset Formula $P^{(0)}\coloneqq P$
\end_inset
y
-\begin_inset Formula $P^{(n+1)}:=P^{(n)\prime}$
+\begin_inset Formula $P^{(n+1)}\coloneqq P^{(n)\prime}$
\end_inset
.
@@ -1605,7 +1605,7 @@ Como
\end_inset
no es cero ni unidad,
-\begin_inset Formula $n:=\text{gr}f>0$
+\begin_inset Formula $n\coloneqq \text{gr}f>0$
\end_inset
, y como el coeficiente principal de
@@ -1634,7 +1634,7 @@ Como
\end_deeper
\begin_layout Enumerate
Si
-\begin_inset Formula $p:=\text{car}K\neq0$
+\begin_inset Formula $p\coloneqq \text{car}K\neq0$
\end_inset
,
@@ -1713,7 +1713,7 @@ Para
\end_inset
y, sea
-\begin_inset Formula $g:=\sum_{j}b_{j}X^{j}$
+\begin_inset Formula $g\coloneqq \sum_{j}b_{j}X^{j}$
\end_inset
,
@@ -1875,7 +1875,7 @@ teorema
\end_inset
],
-\begin_inset Formula $c(p):=\{x\mid x=\text{mcd}_{k\geq0}p_{k}\}$
+\begin_inset Formula $c(p)\coloneqq \{x\mid x=\text{mcd}_{k\geq0}p_{k}\}$
\end_inset
, y [...] si
@@ -2003,11 +2003,11 @@ Si
\end_inset
,
-\begin_inset Formula $f:=\sum_{k}a_{k}X^{k}\in D[X]$
+\begin_inset Formula $f\coloneqq \sum_{k}a_{k}X^{k}\in D[X]$
\end_inset
y
-\begin_inset Formula $n:=\text{gr}(f)$
+\begin_inset Formula $n\coloneqq \text{gr}(f)$
\end_inset
, todas las raíces de
@@ -2043,11 +2043,11 @@ Criterio de reducción:
\end_inset
es primo,
-\begin_inset Formula $f:=\sum_{k}a_{k}X^{k}\in\mathbb{Z}[X]$
+\begin_inset Formula $f\coloneqq \sum_{k}a_{k}X^{k}\in\mathbb{Z}[X]$
\end_inset
es primitivo,
-\begin_inset Formula $n:=\text{gr}(f)$
+\begin_inset Formula $n\coloneqq \text{gr}(f)$
\end_inset
,
@@ -2083,11 +2083,11 @@ Criterio de Eisenstein:
\end_inset
un DFU,
-\begin_inset Formula $f:=\sum_{k}a_{k}X^{k}\in D[X]$
+\begin_inset Formula $f\coloneqq \sum_{k}a_{k}X^{k}\in D[X]$
\end_inset
primitivo y
-\begin_inset Formula $n:=\text{gr}f$
+\begin_inset Formula $n\coloneqq \text{gr}f$
\end_inset
, si existe un irreducible
@@ -2157,7 +2157,7 @@ La irreducibilidad se conserva por automorfismos de dominios, por lo que
\end_inset
,
-\begin_inset Formula $f:=X^{6}+X^{3}+1$
+\begin_inset Formula $f\coloneqq X^{6}+X^{3}+1$
\end_inset
es irreducible, pues
@@ -2181,7 +2181,7 @@ Si
\end_inset
es primo,
-\begin_inset Formula $f(X):=\frac{X^{p}-1}{X-1}=X^{p-1}+X^{p-2}+\dots+X+1$
+\begin_inset Formula $f(X)\coloneqq \frac{X^{p}-1}{X-1}=X^{p-1}+X^{p-2}+\dots+X+1$
\end_inset
es irreducible en
@@ -2275,7 +2275,7 @@ recíproco
\end_inset
son los ceros de
-\begin_inset Formula $f(x):=p(x)/x^{n/2}:K^{*}\to K$
+\begin_inset Formula $f(x)\coloneqq p(x)/x^{n/2}:K^{*}\to K$
\end_inset
, que será de la forma
@@ -2287,12 +2287,12 @@ 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 $k:=n/2$
+\begin_inset Formula $k\coloneqq n/2$
\end_inset
.
Haciendo el cambio de variable
-\begin_inset Formula $y:=x+x^{-1}$
+\begin_inset Formula $y\coloneqq x+x^{-1}$
\end_inset
nos queda una función polinómica de grado
@@ -2681,11 +2681,11 @@ primitiva
\begin_layout Standard
Dado
-\begin_inset Formula $f:=Y^{3}+3pY+2q\in\mathbb{C}[X]$
+\begin_inset Formula $f\coloneqq Y^{3}+3pY+2q\in\mathbb{C}[X]$
\end_inset
, si
-\begin_inset Formula $\omega:=e^{2\pi i/3}$
+\begin_inset Formula $\omega\coloneqq e^{2\pi i/3}$
\end_inset
, existe
@@ -2878,7 +2878,7 @@ Para
\begin_layout Standard
Si
-\begin_inset Formula $f:=aX^{3}+bX^{2}+cX+d\in\mathbb{C}[X]$
+\begin_inset Formula $f\coloneqq aX^{3}+bX^{2}+cX+d\in\mathbb{C}[X]$
\end_inset
, podemos obtener las raíces de
@@ -3078,7 +3078,7 @@ evaluación
\end_inset
viene dado por
-\begin_inset Formula $p(b_{1},\dots,b_{n}):=S(p):=\sum_{i\in\mathbb{N}^{n}}p_{i}b_{1}^{i_{1}}\cdots b_{n}^{i_{n}}$
+\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
[, y
@@ -3098,7 +3098,7 @@ valor
\end_inset
en
-\begin_inset Formula $b:=(b_{1},\dots,b_{n})$
+\begin_inset Formula $b\coloneqq (b_{1},\dots,b_{n})$
\end_inset
].
@@ -3212,11 +3212,11 @@ A[b_{1},\dots,b_{n}]\cong\frac{A[X_{1},\dots,X_{n}]}{\ker S},
\begin_layout Standard
Por ejemplo,
-\begin_inset Formula $b_{1}:=1/\pi$
+\begin_inset Formula $b_{1}\coloneqq 1/\pi$
\end_inset
y
-\begin_inset Formula $b_{2}:=1+\sqrt{\pi}$
+\begin_inset Formula $b_{2}\coloneqq 1+\sqrt{\pi}$
\end_inset
son algebraicamente dependientes, pues satisfaces
@@ -3346,7 +3346,7 @@ Sean
\end_inset
con inversa
-\begin_inset Formula $\tau:=\sigma^{-1}$
+\begin_inset Formula $\tau\coloneqq \sigma^{-1}$
\end_inset
, tomando
@@ -3371,7 +3371,7 @@ Sean
que permuta las indeterminadas.
[Llamamos
-\begin_inset Formula $f^{\sigma}:=\hat{\sigma}(f)$
+\begin_inset Formula $f^{\sigma}\coloneqq \hat{\sigma}(f)$
\end_inset
.]
@@ -3413,7 +3413,7 @@ Todo homomorfismo de anillos conmutativos
\end_inset
dado por
-\begin_inset Formula $\hat{f}(p):=\sum_{i\in\mathbb{N}^{n}}f(p_{i})X_{1}^{i_{1}}\cdots X_{n}^{i_{n}}$
+\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
.
@@ -3859,11 +3859,11 @@ Demostración:
y el resultado se sigue por inducción.
Sean
-\begin_inset Formula $f:=\sum_{i\in\mathbb{N}^{n}}a_{i}X_{1}^{i_{1}}\cdots X_{n}^{i_{n}}$
+\begin_inset Formula $f\coloneqq \sum_{i\in\mathbb{N}^{n}}a_{i}X_{1}^{i_{1}}\cdots X_{n}^{i_{n}}$
\end_inset
y
-\begin_inset Formula $g:=\sum_{i\in\mathbb{N}^{n}}b_{i}X_{1}^{i_{1}}\cdots X_{n}^{i_{n}}$
+\begin_inset Formula $g\coloneqq \sum_{i\in\mathbb{N}^{n}}b_{i}X_{1}^{i_{1}}\cdots X_{n}^{i_{n}}$
\end_inset
, entonces
@@ -3967,19 +3967,19 @@ Queremos ver que, para
.
Con esto, sean
-\begin_inset Formula $A:=\{i\in\mathbb{N}^{n}\mid a_{i}\neq0\}$
+\begin_inset Formula $A\coloneqq \{i\in\mathbb{N}^{n}\mid a_{i}\neq0\}$
\end_inset
,
-\begin_inset Formula $B:=\{j\in\mathbb{N}^{n}\mid b_{j}\neq0\}$
+\begin_inset Formula $B\coloneqq \{j\in\mathbb{N}^{n}\mid b_{j}\neq0\}$
\end_inset
,
-\begin_inset Formula $i^{*}:=\max A$
+\begin_inset Formula $i^{*}\coloneqq \max A$
\end_inset
y
-\begin_inset Formula $j^{*}:=\max B$
+\begin_inset Formula $j^{*}\coloneqq \max B$
\end_inset
, para
generated by cgit v1.2.3 (git 2.46.0) at 2025-10-27 02:39:17 +0000