diff options
Diffstat (limited to 'ealg/n2.lyx')
| -rw-r--r-- | ealg/n2.lyx | 78 |
1 files changed, 39 insertions, 39 deletions
diff --git a/ealg/n2.lyx b/ealg/n2.lyx index cbcd97d..14447e8 100644 --- a/ealg/n2.lyx +++ b/ealg/n2.lyx @@ -221,7 +221,7 @@ Algunas extensiones son \end_inset , entonces -\begin_inset Formula $c:=(a+b\sqrt{m})(a-b\sqrt{m})=a^{2}-mb^{2}\in\mathbb{Q}\setminus0$ +\begin_inset Formula $c\coloneqq (a+b\sqrt{m})(a-b\sqrt{m})=a^{2}-mb^{2}\in\mathbb{Q}\setminus0$ \end_inset , pues @@ -311,7 +311,7 @@ subanillo primo \end_inset a -\begin_inset Formula $\mathbb{Z}1:=\{n1_{A}\}_{n\in\mathbb{Z}}$ +\begin_inset Formula $\mathbb{Z}1\coloneqq \{n1_{A}\}_{n\in\mathbb{Z}}$ \end_inset , el menor subanillo de @@ -389,7 +389,7 @@ Demostración: \end_inset dado por -\begin_inset Formula $f(n):=n1$ +\begin_inset Formula $f(n)\coloneqq n1$ \end_inset es un homomorfismo inyectivo y la propiedad universal nos da un homomorfismo @@ -403,7 +403,7 @@ Demostración: . Es claro entonces que -\begin_inset Formula $K':=\tilde{f}(\mathbb{Q})$ +\begin_inset Formula $K'\coloneqq \tilde{f}(\mathbb{Q})$ \end_inset es isomorfo a @@ -506,7 +506,7 @@ grado \end_inset a -\begin_inset Formula $[L:K]:=\dim_{K}L$ +\begin_inset Formula $[L:K]\coloneqq \dim_{K}L$ \end_inset , la dimensión de @@ -926,7 +926,7 @@ status open \begin_layout Plain Layout Sean -\begin_inset Formula $a:=\sum_{i\in I}a_{i}\alpha_{1}^{i_{1}}\cdots\alpha_{p}^{i_{p}},b:=\sum_{j\in J}b_{i}\beta_{1}^{j_{1}}\cdots\beta_{q}^{i_{q}}\in K(S)$ +\begin_inset Formula $a\coloneqq \sum_{i\in I}a_{i}\alpha_{1}^{i_{1}}\cdots\alpha_{p}^{i_{p}},b\coloneqq \sum_{j\in J}b_{i}\beta_{1}^{j_{1}}\cdots\beta_{q}^{i_{q}}\in K(S)$ \end_inset , con @@ -1586,7 +1586,7 @@ Extensiones \end_inset dada por -\begin_inset Formula $\hat{\sigma}(\tau):=\sigma\circ\tau\circ\sigma^{-1}$ +\begin_inset Formula $\hat{\sigma}(\tau)\coloneqq \sigma\circ\tau\circ\sigma^{-1}$ \end_inset es un isomorfismo de grupos, dado que @@ -1709,15 +1709,15 @@ status open \begin_layout Plain Layout Para esto se usa el transporte de estructuras. Sea -\begin_inset Formula $\varphi:K\to(L_{0}:=K[X]/(g))$ +\begin_inset Formula $\varphi:K\to(L_{0}\coloneqq K[X]/(g))$ \end_inset el homomorfismo -\begin_inset Formula $\varphi(a):=a+(g)$ +\begin_inset Formula $\varphi(a)\coloneqq a+(g)$ \end_inset , definimos -\begin_inset Formula $L:=K\amalg(L_{0}\setminus\varphi(K))$ +\begin_inset Formula $L\coloneqq K\amalg(L_{0}\setminus\varphi(K))$ \end_inset y las operaciones en @@ -1725,11 +1725,11 @@ Para esto se usa el transporte de estructuras. \end_inset -\begin_inset Formula $a+b:=\psi^{-1}(\psi(a)+\psi(b))$ +\begin_inset Formula $a+b\coloneqq \psi^{-1}(\psi(a)+\psi(b))$ \end_inset y -\begin_inset Formula $ab:=(\psi(a)\psi(b))$ +\begin_inset Formula $ab\coloneqq (\psi(a)\psi(b))$ \end_inset , donde @@ -1737,7 +1737,7 @@ Para esto se usa el transporte de estructuras. \end_inset viene dado por -\begin_inset Formula $\psi(a):=\varphi(a)$ +\begin_inset Formula $\psi(a)\coloneqq \varphi(a)$ \end_inset para @@ -1745,7 +1745,7 @@ Para esto se usa el transporte de estructuras. \end_inset y -\begin_inset Formula $\psi(a):=a$ +\begin_inset Formula $\psi(a)\coloneqq a$ \end_inset para @@ -1812,7 +1812,7 @@ Demostración: \end_inset , -\begin_inset Formula $L:=K[X]/(g)$ +\begin_inset Formula $L\coloneqq K[X]/(g)$ \end_inset es un cuerpo. @@ -2092,7 +2092,7 @@ Si \end_inset y -\begin_inset Formula $\omega:=e^{2\pi i/n}\in\mathbb{C}$ +\begin_inset Formula $\omega\coloneqq e^{2\pi i/n}\in\mathbb{C}$ \end_inset , @@ -2315,7 +2315,7 @@ simple \end_inset y -\begin_inset Formula $\alpha:=[X]=X+I\in L$ +\begin_inset Formula $\alpha\coloneqq [X]=X+I\in L$ \end_inset la raíz, para @@ -2501,7 +2501,7 @@ Como \end_deeper \begin_layout Enumerate Sea -\begin_inset Formula $n:=\text{gr}f$ +\begin_inset Formula $n\coloneqq \text{gr}f$ \end_inset , @@ -2542,7 +2542,7 @@ Si \end_inset pero entonces -\begin_inset Formula $g:=\sum_{i=0}^{n-1}a_{i}X^{i}\in K[X]\setminus0$ +\begin_inset Formula $g\coloneqq \sum_{i=0}^{n-1}a_{i}X^{i}\in K[X]\setminus0$ \end_inset tendría a @@ -2708,7 +2708,7 @@ Si \end_inset es primo y -\begin_inset Formula $\xi:=e^{2\pi i/p}$ +\begin_inset Formula $\xi\coloneqq e^{2\pi i/p}$ \end_inset , @@ -2965,11 +2965,11 @@ Demostración: . Sean -\begin_inset Formula $F_{p}:=\mathbb{Q}[\sqrt{p}]$ +\begin_inset Formula $F_{p}\coloneqq \mathbb{Q}[\sqrt{p}]$ \end_inset y -\begin_inset Formula $F_{q}:=\mathbb{Q}[\sqrt{q}]$ +\begin_inset Formula $F_{q}\coloneqq \mathbb{Q}[\sqrt{q}]$ \end_inset , @@ -2986,12 +2986,12 @@ Demostración: . Claramente -\begin_inset Formula $S:=\{a+b\sqrt{p}+c\sqrt{q}+d\sqrt{pq}\}_{a,b,c,d\in\mathbb{Q}}\subseteq F_{p}F_{q}$ +\begin_inset Formula $S\coloneqq \{a+b\sqrt{p}+c\sqrt{q}+d\sqrt{pq}\}_{a,b,c,d\in\mathbb{Q}}\subseteq F_{p}F_{q}$ \end_inset . Sea ahora -\begin_inset Formula $\alpha:=\sqrt{p}+\sqrt{q}\in S$ +\begin_inset Formula $\alpha\coloneqq \sqrt{p}+\sqrt{q}\in S$ \end_inset , @@ -3105,7 +3105,7 @@ status open \end_inset , es algebraico, y si -\begin_inset Formula $d:=\text{Irr}(\alpha,K)$ +\begin_inset Formula $d\coloneqq \text{Irr}(\alpha,K)$ \end_inset , @@ -3154,7 +3154,7 @@ Sean \end_inset un isomorfismo de cuerpos y -\begin_inset Formula $f':=\sigma(f)$ +\begin_inset Formula $f'\coloneqq \sigma(f)$ \end_inset con una raíz @@ -3213,7 +3213,7 @@ Demostración: . Sea -\begin_inset Formula $\hat{\sigma}(g(\alpha)):=\sigma(g)(\alpha')$ +\begin_inset Formula $\hat{\sigma}(g(\alpha))\coloneqq \sigma(g)(\alpha')$ \end_inset , @@ -3415,7 +3415,7 @@ Sea \end_inset Sea -\begin_inset Formula $f:=\text{Irr}(\alpha,K)$ +\begin_inset Formula $f\coloneqq \text{Irr}(\alpha,K)$ \end_inset , sabemos que al ser @@ -3880,11 +3880,11 @@ grupo cíclico \end_inset a -\begin_inset Formula $C_{n}:=\{1,a,a^{2},\dots,a^{n-1}\}$ +\begin_inset Formula $C_{n}\coloneqq \{1,a,a^{2},\dots,a^{n-1}\}$ \end_inset con [...] -\begin_inset Formula $a^{i}a^{j}:=a^{[i+j]_{n}}$ +\begin_inset Formula $a^{i}a^{j}\coloneqq a^{[i+j]_{n}}$ \end_inset [...]. @@ -3921,7 +3921,7 @@ D_{n}:=\{1,a,a^{2},\dots,a^{n-1},b,ab,a^{2}b,\dots,a^{n-1}b\} \end_inset con la operación -\begin_inset Formula $(a^{i_{1}}b^{j_{1}})(a^{i_{2}}b^{j_{2}}):=a^{[i_{1}+(-1)^{j_{1}}i_{2}]_{n}}b^{[j_{1}+j_{2}]_{2}}$ +\begin_inset Formula $(a^{i_{1}}b^{j_{1}})(a^{i_{2}}b^{j_{2}})\coloneqq a^{[i_{1}+(-1)^{j_{1}}i_{2}]_{n}}b^{[j_{1}+j_{2}]_{2}}$ \end_inset . @@ -3962,7 +3962,7 @@ Teorema de Lagrange: grupo alternado \series default [...] a -\begin_inset Formula $A_{n}:=\ker\text{sgn}$ +\begin_inset Formula $A_{n}\coloneqq \ker\text{sgn}$ \end_inset , el subgrupo de @@ -4227,7 +4227,7 @@ status open Demostración: \series default Llamando -\begin_inset Formula $m:=\text{Exp}K^{*}$ +\begin_inset Formula $m\coloneqq \text{Exp}K^{*}$ \end_inset , @@ -4293,7 +4293,7 @@ Llamando . Entonces el orden de -\begin_inset Formula $a:=a_{1}+\dots+a_{k}$ +\begin_inset Formula $a\coloneqq a_{1}+\dots+a_{k}$ \end_inset es @@ -4318,7 +4318,7 @@ Sean \end_inset primo y -\begin_inset Formula $\xi:=e^{2\pi i/p}$ +\begin_inset Formula $\xi\coloneqq e^{2\pi i/p}$ \end_inset , @@ -4363,7 +4363,7 @@ Demostración: \end_inset donde -\begin_inset Formula $\sigma_{k}(\xi):=\xi^{k}$ +\begin_inset Formula $\sigma_{k}(\xi)\coloneqq \xi^{k}$ \end_inset . @@ -4728,7 +4728,7 @@ cuerpo de números algebraicos cuerpo de los números algebraicos \series default a -\begin_inset Formula ${\cal A}:=\overline{\mathbb{Q}}_{\mathbb{C}}$ +\begin_inset Formula ${\cal A}\coloneqq \overline{\mathbb{Q}}_{\mathbb{C}}$ \end_inset , y @@ -4767,7 +4767,7 @@ números algebraicos \end_inset para -\begin_inset Formula $\xi_{p}:=e^{2\pi i/p}$ +\begin_inset Formula $\xi_{p}\coloneqq e^{2\pi i/p}$ \end_inset , pero @@ -4935,7 +4935,7 @@ Para \end_inset , existe -\begin_inset Formula $f:=\sum_{i=0}^{n}a_{i}X^{i}\in L[X]$ +\begin_inset Formula $f\coloneqq \sum_{i=0}^{n}a_{i}X^{i}\in L[X]$ \end_inset que tiene a |