aboutsummaryrefslogtreecommitdiff
path: root/ealg/n2.lyx
blob: e3050d930287a32fd0a2bbe1051a8d53149673c5 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
#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 Standard
Si 
\begin_inset Formula $K$
\end_inset

 es un cuerpo y 
\begin_inset Formula $A$
\end_inset

 es un anillo, todo homomorfismo de anillos 
\begin_inset Formula $h:K\to A$
\end_inset

 es un isomorfismo 
\begin_inset Formula $K\to h(K)$
\end_inset

.
 En efecto, 
\begin_inset Formula $\ker h\subseteq K$
\end_inset

 es un ideal y los únicos ideales de 
\begin_inset Formula $K$
\end_inset

 son 0 y 
\begin_inset Formula $K$
\end_inset

, pero 
\begin_inset Formula $1\notin\ker h$
\end_inset

, luego 
\begin_inset Formula $\ker h=0$
\end_inset

, 
\begin_inset Formula $h$
\end_inset

 es inyectivo y 
\begin_inset Formula $h:K\to h(K)$
\end_inset

 es biyectivo.
\end_layout

\begin_layout Standard
Una 
\series bold
extensión
\series default
 (
\series bold
de cuerpos
\series default
) es un par de cuerpos 
\begin_inset Formula $(K,L)$
\end_inset

 con 
\begin_inset Formula $K$
\end_inset

 es subcuerpo de 
\begin_inset Formula $L$
\end_inset

, que representamos como 
\begin_inset Formula $K\subset L$
\end_inset

, 
\begin_inset Formula $L/K$
\end_inset

 o, si 
\begin_inset Formula $K\neq L$
\end_inset

, como 
\begin_inset Formula $K\subsetneq L$
\end_inset

.
\end_layout

\begin_layout Standard
Algunas extensiones son 
\begin_inset Formula $\mathbb{Q}\subset\mathbb{R}$
\end_inset

, 
\begin_inset Formula $\mathbb{R}\subset\mathbb{C}$
\end_inset

 y, para todo cuerpo 
\begin_inset Formula $K$
\end_inset

, 
\begin_inset Formula $K\subset K(X)$
\end_inset

, donde 
\begin_inset Formula $K(X)$
\end_inset

 es el cuerpo de fracciones de 
\begin_inset Formula $K[X]$
\end_inset

.
 Otras son 
\begin_inset Formula $\mathbb{Q}\subset\mathbb{Q}[\sqrt{m}]$
\end_inset

 para 
\begin_inset Formula $m\in\mathbb{Z}$
\end_inset

 no cuadrado y 
\begin_inset Formula $\mathbb{Q}\subset\mathbb{Q}[i]$
\end_inset

.
 En ef
\begin_inset Note Note
status open

\begin_layout Plain Layout
a.pdf::27
\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
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
eremember
\end_layout

\end_inset


\end_layout

\end_body
\end_document