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¤U¤@­¶: Roots of Polynomials ¤W¤@­¶: ¤¤¯Å Field ªº©Ê½è «e¤@­¶: Algebraic Elements

Algebraic Closure

·í F ¬O¤@­Ó field, L ¬O F ªº¤@­Ó extension ®É, §Ú­Ì¥i¥H±N L ¤¤ªº¤¸¯À¤À¦¨ algebraic over F ©M¤£¬O algebraic over F ªº¨âºØ. ¦b³o¤@¸`¤¤§Ú­Ì±N±´°Q L ¤¤©Ò¦³ algebraic over F ªº¤¸¯À©Ò¦¨¤§¶°¦X.

Definition 10.2.1   °²³] F ¬O¤@­Ó field ¥B L ¬O F ªº¤@­Ó extension. §Ú­Ì¥O

$\displaystyle \overline{L_F}$ = {a $\displaystyle \in$ L | a ¬O algebraic over F},

ºÙ¤§¬° F ¦b L ªº algebraic closure.

F ¤¤ªº¤¸¯À·íµM¬O algebraic over F, ©Ò¥H¥Ñ©w¸qª¾ F $ \subseteq$ $ \overline{L_F}$ $ \subseteq$ L. ¥t¥ ¦pªG L ¬O F ªº¤@­Ó finite extension, «h¥Ñ Lemma 9.4.5 ª¾ L ¤¤ªº¤¸¯À³£ algebraic over F, ©Ò¥H¦b³o­Ó°²³]¤§¤U $ \overline{L_F}$ = L.

±µ¤U¨Ó§Ú­Ì­nÃÒ©ú $ \overline{L_F}$ ªº¤@­Ó­«­n©Ê½è, §Y $ \overline{L_F}$ ¬O¤@­Ó field. ´«¨¥¤§, §Ú­Ì­nÃÒ©ú­Y a, b $ \in$ $ \overline{L_F}$, ¨ä¤¤ b$ \ne$ 0, «h a - b ¥H¤Î a . b-1 ¬Ò¦b $ \overline{L_F}$ ¤¤ (Lemma 9.1.4). ­n¦p¦óÃÒ©ú³o¨Ç¤¸¯À³£¬O algebraic over F ©O? ·íµM¤£¥i¯à¥Î§ä polynomial ªº¤èªk, §Ú­Ì¥²¶·ÂǧU Theorem 10.1.9. ¦b³o¤§«e§Ú­Ì¥ý±À¼s¤@¤U Definition 10.1.6.

Definition 10.2.2   °²³] F ¬O¤@­Ó field ¥B L ¬O F ªº¤@­Ó extension. ­Y a1,..., an $ \in$ L, «h©w F(a1,..., an) ªí¥Ü¬° L ¤¤¥]§t F ¥H¤Î a1,..., an ³Ì¤pªº field.

Lemma 10.2.3   °²³] F ¬O¤@­Ó field ¥B L ¬O F ªº¤@­Ó extension. ­Y a1,..., an $ \in$ L ¬Ò¬° algebraic over F, «h F(a1,..., an) ¬O F ªº¤@­Ó finite extension. ¨Æ¹ê¤W, ¦pªG¤wª¾ a1,..., an over F ªº degree ¤À§O¬° m1,..., mn, «h

[F(a1,..., an) : F]$\displaystyle \le$m1 ... mn.

µý ©ú. ¬°¤F¤è«K, §Ú­Ì¥O

K1 = F(a1), K2 = K1(a2) = F(a1, a2),..., Kn = Kn - 1(an) = F(a1,..., an).

¹ï¥ô·Nªº i, §Ú­Ì¦³ [Ki : Ki - 1] = [Ki - 1(ai) : Ki]$ \le$mi. ³o¸Ì [Ki - 1(ai) : Ki - 1] ·|¤p©ó©Îµ¥©ó mi ªº­ì¦]¬O: ¥Ñ Corollary 10.1.7 ª¾ [Ki - 1(ai) : Ki - 1] ªº­È­è¦n¬O ai over Ki - 1 ªº minimal polynomial qi(x) $ \in$ Ki - 1[x] ªº degree. µM¦Ó¥Ñ°²³] ai over F ªº minimal polynomial pi(x) $ \in$ F[x] ªº degree ¬° mi. ¥Ñ©ó pi(x) $ \in$ F[x] $ \subseteq$ Ki - 1[x] ¥B pi(ai) = 0, ¬G¥Ñ qi(x) ¬O ai over Ki - 1 ªº minimal polynomial ªº°²³]ª¾ deg(qi(x))$ \le$deg(pi(x)) = mi. ¬Gª¾

[Ki : Ki - 1] = [Ki - 1(ai) : Ki - 1] = deg(qi(x))$\displaystyle \le$mi.

²¦b¥Ñ©ó¨C¤@¬q [Ki : Ki - 1] ³£¬O¦³­­ªº, ©Ò¥H§Ú­Ì¥i¥H³sÄò®M¥Î Theorem 9.4.6 ±o:

[F(a1,..., an) : F] = [Kn : Kn - 1][Kn - 1 : F]  
  = [Kn : Kn - 1][Kn - 1 : Kn - 2][Kn - 2 : F]  
  $\displaystyle \vdots$    
  = [Kn : Kn - 1] ... [K1 : F]$\displaystyle \le$mn ... m1.  

¬G±oÃÒ F(a1,..., an) ¬O F ªº¤@­Ó finite extension. $ \qedsymbol$

§Q¥Î Lemma 10.2.3 §Ú­Ì°¨¤W¥i±oª¾ $ \overline{L_F}$ ¬O¤@­Ó field.

Theorem 10.2.4   °²³] F ¬O¤@­Ó field ¥B L ¬O F ªº¤@­Ó extension. ­Y a, b $ \in$ L, ¨ä¤¤ b$ \ne$ 0, ¬Ò¬° algebraic over F, «h a + b, a - b, a . b ¥H¤Î a . b-1 ¬Ò¬° algebraic over F. ¥Ñ¦¹§Ú­Ì¥i±o $ \overline{L_F}$ ¬O¤@­Ó field.

µý ©ú. ¥Ñ Lemma 10.2.3 §Ú­Ìª¾ F(a, b) ¬O F ªº¤@­Ó finite extension. ¥Ñ©ó a, b $ \in$ F(a, b), b$ \ne$ 0 ¥B F(a, b) ¬O¤@­Ó field, §Ú­Ì¦ÛµM¦³ a + b, a - b, a . b ¥H¤Î a . b-1 ¬Ò¬° F(a, b) ªº¤¸¯À. ¬G¥Ñ Theorem 10.1.9 (©Î Lemma 10.1.3) ª¾³o¥|­Ó¤¸¯À¬Ò¬° algebraic over F.

¤µ­Y a, b $ \in$ $ \overline{L_F}$, ¨ä¤¤ b$ \ne$ 0, «h¥Ñ©w¸qª¾ a, b ¬Ò¬° algebraic over F. ¬G¥Ñ«eª¾ a + b, a - b, a . b ¥H¤Î a . b-1 ¬Ò¬° algebraic over F. ¬Gª¾³o¥|­Ó¤¸¯À¬Ò¦b $ \overline{L_F}$ ¤¤, ¦]¦¹±oÃÒ $ \overline{L_F}$ ¬O¤@­Ó field. $ \qedsymbol$

°²³] L ¬O F ªº¤@­Ó extension, ¥B K ¬O L over F ªº subextension (§Y F $ \subseteq$ K $ \subseteq$ L). L ¤¤¬O algebraic over K ªº¤¸¯À¥¼¥²¬O algebraic over F. ¤£¹L L ¤¤¬O algebraic over F ªº¤¸¯À´N¤@©w¬O algebraic over K. ³o¬O¦]¬°­Y a $ \in$ $ \overline{L_F}$ (§Y a $ \in$ L ¬O algebraic over F), ªí¥Ü¦b F[x] ¤¤¦s¦b f (x)$ \ne$ 0 ¨Ï±o f (a) = 0. ¥Ñ©ó f (x) $ \in$ F[x] $ \subseteq$ K[x], §Ú­Ì¦ÛµM±o a ¤]¬O algebraic over K. ¬G±o a $ \in$ $ \overline{L_K}$, ´«¥y¸Ü»¡§Ú­ÌÁ`¬O¦³

$\displaystyle \overline{L_F}$ $\displaystyle \subseteq$ $\displaystyle \overline{L_K}$.

§Ú­Ì¦³¿³½ìª¾¹D¤°»ò®É­Ô $ \overline{L_F}$ ·|µ¥©ó $ \overline{L_K}$. ¥H¤U¬O¤@­Ó¨Ò¤l.

Lemma 10.2.5   °²³] F ¬O¤@­Ó field, L ¬O F ªº¤@­Ó extension, ¥B K ¬O L over F ªº subextension. ­Y K ¬O F ªº¤@­Ó finite extension, «h $ \overline{L_F}$ = $ \overline{L_K}$

µý ©ú. §Ú­Ì¤wª¾ $ \overline{L_F}$ $ \subseteq$ $ \overline{L_K}$, ©Ò¥H¥u­nÃÒ©ú $ \overline{L_K}$ $ \subseteq$ $ \overline{L_F}$. ¤]´N¬O­nÃÒ©ú: ­Y a $ \in$ L ¬O algebraic over K, «h a ¬O algebraic over F. §Ú­Ì¦Ò¼ K(a) ³o¤@­Ó field. ¥Ñ°²³] a ¬O algebraic over K, ¬G§Q¥Î Corollary 10.1.7 ª¾ K(a) ¬O K ªº¤@­Ó finite extension. ¦A¥[¤W K ¬O F ªº¤@­Ó finite extension, ®M¥Î Theorem 9.4.6 ¥i±o

[K(a) : F] = [K(a) : K][K : F],

¦]¦¹ K(a) ¬O F ªº¤@­Ó finite extension. ¬G§Q¥Î a $ \in$ K(a) ¥H¤Î Theorem 10.1.9 (©Î Lemma 10.1.3) ª¾ a ¬O algebraic over F. $ \qedsymbol$

§Ú­Ì¥i¥H±N Lemma 10.2.5 ±À¼s¨ì§ó¤@¯ëªºª¬ªp. ¦^ÅU¤@¤U­Y K ¬O F ªº¤@­Ó algebraic extension ªí¥Ü K ¤¤ªº¤¸¯À¬Ò¬° algebraic over F. ¦b Lemma 10.2.5 ¤¤ªº°²³] K ¬O F ªº finite extension, ©Ò¥H¦ÛµM¬O F ªº¤@­Ó algebraic extension. §Ú­Ì­n±N Lemma 10.2.5 ±À¼s¨ì K ¬O F ªº algebraic extension ³o­Óª¬ªp.

Theorem 10.2.6   °²³] F ¬O¤@­Ó field, L ¬O F ªº¤@­Ó extension, ¥B K ¬O L over F ªº subextension. ­Y K ¬O F ªº¤@­Ó algebraic extension, «h $ \overline{L_F}$ = $ \overline{L_K}$

µý ©ú. ©M Lemma 10.2.5 ¬Û¦Pªº±¡§Î, §Ú­Ì¥u­nÃÒ©ú: ­Y a $ \in$ L ¬O algebraic over K, «h a ¬O algebraic over F. ¤£¹L³o¸Ì¸I¨ìªºª¬ªp¬O K ¥i¯à¤£¬O finite extension over F, ©Ò¥H§Ú­Ì¤£¯àª½±µ®M¥Î Lemma 10.2.5. ­n§JªA³o­Ó§xÃø, §Ú­Ì¥²¶··Q¿ìªk§ä¨ì¤@­Ó F ªº finite extension K' ¥Bº¡¨¬ a ¬O algebraic over K'. ¦p¦¹¦A®M¥Î Lemma 10.2.5 ±oÃÒ a ¬O algebraic over F.

¥Ñ°²³] a ¬O algebraic over K, ª¾¦s¦b f (x)$ \ne$ 0 ¥B f (x) $ \in$ K[x] ¨Ï±o f (a) = 0. °²³] f (x) = anxn + ... + a0. ¥Ñ©ó an,..., a0 $ \in$ K ¥B K ¬O F ªº¤@­Ó algebraic extension, ¬Gª¾ an,..., a0 ¬Ò¬° algebraic over F. ¥O K' = F(an,..., a0), ¥Ñ Lemma 10.2.3 ª¾ K' ¬O F ªº¤@­Ó finite extension. ¬G§Q¥Î Lemma 10.2.5 ª¾ $ \overline{L_{K'}}$ = $ \overline{L_F}$. ¥t¥ ¥Ñ©ó an,..., a0 $ \in$ F(an,..., a0) = K', §Ú­Ìª¾ f (x) $ \in$ K'[x]. ¬G¥Ñ f (a) = 0 ª¾ a ¬O algebraic over K'. ´«¨¥¤§, §Ú­Ì¦³ a $ \in$ $ \overline{L_{K'}}$, ¬G¥Ñ $ \overline{L_{K'}}$ = $ \overline{L_F}$ ±oª¾ a $ \in$ $ \overline{L_F}$. ¦]¦¹±oÃÒ a ¬O algebraic over F. $ \qedsymbol$

§Ú­Ì¤wª¾ $ \overline{L_F}$ ¬O¤@­Ó filed (Theorem 10.2.4) ¥B F $ \subseteq$ $ \overline{L_F}$ $ \subseteq$ L. ¦pªG§Ú­Ì¦A¦¬¶° L ¤¤¬O algebraic over $ \overline{L_F}$ ªº¤¸¯À·|¤£·|±o¨ì§ó¤jªº field ªº©O? ´«¥y¸Ü¨Ó»¡, §Ú­Ì·Qª¾¹D $ \overline{L_{\overline{L_F}}}$ (¤£­n³Q³o²Å¸¹À µÛ¤F) ¬O¤°»ò? ¨Æ¹ê¤W©Ò¿×ªº algebraic closure ´N¬O»¡ L ¤¤ algebraic over $ \overline{L_F}$ ªº¤¸¯À©Ò¦¨ªº¶°¦X´N¬O $ \overline{L_F}$ ¦Û¤v.

Corollary 10.2.7   °²³] F ¬O¤@­Ó field ¥B L ¬O F ªº¤@­Ó extension, ­Y a $ \in$ L ¥B a ¬O algebraic over $ \overline{L_F}$, «h a ¬O algebraic over F. ¤]´N¬O»¡, §Ú­Ì¦³

$\displaystyle \overline{L_{\overline{L_F}}}$ = $\displaystyle \overline{L_F}$.

µý ©ú. ­º¥ýª`·N¥Ñ©w¸q $ \overline{L_F}$ ¤¤ªº¤¸¯À³£¬O algebraic over F, ¬Gª¾ $ \overline{L_F}$ ¬O F ªº¤@­Ó algebraic extension. ¦]¦¹­Y¥O K = $ \overline{L_F}$, «h K ²Å¦X Theorem 10.2.6 ªº±ø¥ó, ¬Gª¾ $ \overline{L_K}$ = $ \overline{L_F}$. ¤]¦]¦¹­Y a $ \in$ L ¬O algebraic over $ \overline{L_F}$ = K, ªí¥Ü a $ \in$ $ \overline{L_K}$. ¬G¥Ñ $ \overline{L_K}$ = $ \overline{L_F}$ ±oª¾ a $ \in$ $ \overline{L_F}$, ¤]´N¬O»¡ a ¬O algebraic over F. $ \qedsymbol$


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¤U¤@­¶: Roots of Polynomials ¤W¤@­¶: ¤¤¯Å Field ªº©Ê½è «e¤@­¶: Algebraic Elements
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