¤U¤@¶: Roots of Polynomials
¤W¤@¶: ¤¤¯Å Field ªº©Ê½è
«e¤@¶: Algebraic Elements
·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¤@¸`¤¤§Ú̱N±´°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
= {
a 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 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
= L.
±µ¤U¨Ó§ÚÌnÃÒ©ú
ªº¤@Ó«n©Ê½è, §Y
¬O¤@Ó
field. ´«¨¥¤§, §ÚÌnÃÒ©úY
a, b , ¨ä¤¤ b 0, «h
a - b ¥H¤Î
a . b-1 ¬Ò¦b
¤¤ (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 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 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]
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, §Ú̦³
[
Ki :
Ki - 1] = [
Ki - 1(
ai) :
Ki]
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)
Ki - 1[
x] ªº degree.
µM¦Ó¥Ñ°²³]
ai over
F ªº minimal polynomial
pi(
x)
F[
x] ªº
degree ¬°
mi. ¥Ñ©ó
pi(
x)
F[
x]
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))
deg(
pi(
x)) =
mi. ¬Gª¾
[
Ki :
Ki - 1] = [
Ki - 1(
ai) :
Ki - 1] = deg(
qi(
x))
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] |
|
|
|
|
|
|
= |
[Kn : Kn - 1] ... [K1 : F]mn ... m1. |
|
¬G±oÃÒ
F(
a1,...,
an) ¬O
F ªº¤@Ó finite
extension.
§Q¥Î Lemma 10.2.3 §ÚÌ°¨¤W¥i±oª¾
¬O¤@Ó field.
Theorem 10.2.4
°²³]
F ¬O¤@Ó field ¥B
L ¬O
F ªº¤@Ó extension. Y
a,
b L,
¨ä¤¤
b 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
¬O¤@Ó field.
µý ©ú.
¥Ñ Lemma
10.2.3 §Ú̪¾
F(
a,
b) ¬O
F ªº¤@Ó finite extension.
¥Ñ©ó
a,
b F(
a,
b),
b 0 ¥B
F(
a,
b) ¬O¤@Ó field, §Ú̦۵M¦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 , ¨ä¤¤ b 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
¤¤,
¦]¦¹±oÃÒ
¬O¤@Ó field.
°²³] L ¬O F ªº¤@Ó extension, ¥B K ¬O L over F ªº
subextension (§Y
F K 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
(§Y a L ¬O algebraic over F), ªí¥Ü¦b F[x] ¤¤¦s¦b f (x) 0 ¨Ï±o f (a) = 0. ¥Ñ©ó
f (x) F[x] K[x], §Ú̦۵M±o a
¤]¬O algebraic over K. ¬G±o
a , ´«¥y¸Ü»¡§ÚÌÁ`¬O¦³
§Ú̦³¿³½ìª¾¹D¤°»ò®ÉÔ
·|µ¥©ó
. ¥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
=
µý ©ú.
§Ṳ́wª¾
, ©Ò¥H¥unÃÒ©ú
. ¤]´N¬OnÃÒ©ú: Y
a L ¬O algebraic
over
K, «h
a ¬O algebraic over
F. §Ú̦Ҽ
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 K(
a) ¥H¤Î Theorem
10.1.9 (©Î Lemma
10.1.3) ª¾
a ¬O
algebraic over
F.
§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
=
µý ©ú.
©M Lemma
10.2.5 ¬Û¦Pªº±¡§Î, §ÚÌ¥unÃÒ©ú: Y
a 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§Ṳ́£¯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) 0 ¥B
f (x) K[x] ¨Ï±o f (a) = 0. °²³]
f (x) = anxn + ... + a0. ¥Ñ©ó
an,..., a0 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
= . ¥t¥ ¥Ñ©ó
an,..., a0 F(an,..., a0) = K', §Ú̪¾
f (x) K'[x]. ¬G¥Ñ
f (a) = 0 ª¾ a ¬O algebraic over K'. ´«¨¥¤§, §Ú̦³
a , ¬G¥Ñ
= ±oª¾
a .
¦]¦¹±oÃÒ a ¬O algebraic over F.
§Ṳ́wª¾
¬O¤@Ó filed (Theorem 10.2.4) ¥B
F L. ¦pªG§Ú̦A¦¬¶° L ¤¤¬O algebraic
over
ªº¤¸¯À·|¤£·|±o¨ì§ó¤jªº field ªº©O? ´«¥y¸Ü¨Ó»¡,
§ÚÌ·Qª¾¹D
(¤£n³Q³o²Å¸¹À µÛ¤F) ¬O¤°»ò?
¨Æ¹ê¤W©Ò¿×ªº algebraic closure ´N¬O»¡ L ¤¤ algebraic over
ªº¤¸¯À©Ò¦¨ªº¶°¦X´N¬O
¦Û¤v.
Corollary 10.2.7
°²³]
F ¬O¤@Ó field ¥B
L ¬O
F ªº¤@Ó extension, Y
a L ¥B
a ¬O algebraic over
, «h
a ¬O algebraic over
F.
¤]´N¬O»¡, §Ú̦³
=
.
µý ©ú.
º¥ýª`·N¥Ñ©w¸q
¤¤ªº¤¸¯À³£¬O algebraic over
F, ¬Gª¾
¬O
F ªº¤@Ó algebraic extension. ¦]¦¹Y¥O
K =
,
«h
K ²Å¦X Theorem
10.2.6 ªº±ø¥ó, ¬Gª¾
=
.
¤]¦]¦¹Y
a L ¬O algebraic over
=
K, ªí¥Ü
a . ¬G¥Ñ
=
±oª¾
a ,
¤]´N¬O»¡
a ¬O algebraic over
F.
¤U¤@¶: Roots of Polynomials
¤W¤@¶: ¤¤¯Å Field ªº©Ê½è
«e¤@¶: Algebraic Elements
Administrator
2005-06-18