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¤U¤@­¶: Euclidean Domain ¤W¤@­¶: Integral Domain ¤Wªº¤À¸Ñ©Ê½è «e¤@­¶: Integral Domain ¤Wªº¤À¸Ñ©Ê½è

Divisor

¦b integral domain ¸Ì¤¸¯Àªº¤À¸Ñ¤j®aÀ³¸Ó³£¤F¸Ñ³Ì°ò¥»ªº¤¸¯À´N¬O irreducible elements ©M prime elements. §Ú­Ì±N¦³¨t²Îªº±´°Q¥¦­Ìªº°ò¥»©Ê½è.

­º¥ý§Ú­ÌÁÙ¬O¹ï¤@­Ó¤¸¯Àªº¦]¼Æµ¹¤@­Ó¥¿¦¡ªº©w¸q.

Definition 8.1.1   ¥O R ¬O¤@­Ó integral domain ¥B a, d $ \in$ R ¬O R ¤¤¨â­Ó¤£¬° 0 ªº¤¸¯À. ¦pªG¦s¦b r $ \in$ R º¡¨¬ a = d . r, «hºÙ d ¬° a ¦b R ¤¤ªº¤@­Ó divisor ¥B°O¬° d | a.

¦^ÅU¤@¤U­Y R ¬O integral domain ¥B d $ \in$ R, «h $ \bigl($d$ \bigr)$ = {d . r | r $ \in$ R} ©Ò¥H¥Ñ¤W¤@­Ó©w¸q§Ú­Ì«Ü®e©öª¾ d | a ­Y¥B°ß­Y a $ \in$ $ \bigl($d$ \bigr)$. µM¦Ó­Y a $ \in$ $ \bigl($d$ \bigr)$, ¥Ñ $ \bigl($d$ \bigr)$ ¬O¤@­Ó ideal ª¾¹ï¥ô·Nªº r $ \in$ R ¬Ò¦³ a . r $ \in$ $ \bigl($d$ \bigr)$. ¬G±o $ \bigl($a$ \bigr)$ $ \subseteq$ $ \bigl($d$ \bigr)$. ¤Ï¤§­Y $ \bigl($a$ \bigr)$ $ \subseteq$ $ \bigl($d$ \bigr)$, ¥Ñ a $ \in$ $ \bigl($a$ \bigr)$ ±oª¾ a $ \in$ $ \bigl($d$ \bigr)$. ´«¥y¸Ü»¡ a $ \in$ $ \bigl($d$ \bigr)$ ­Y¥B°ß­Y $ \bigl($a$ \bigr)$ $ \subseteq$ $ \bigl($d$ \bigr)$, ¦]¦¹§Ú­Ì¦³¥H¤Uªºµ²½×:

Lemma 8.1.2   ¥O R ¬O¤@­Ó integral domain ¥B a, d $ \in$ R $ \setminus$ {0} . «h d | a ­Y¥B°ß­Y $ \bigl($a$ \bigr)$ $ \subseteq$ $ \bigl($d$ \bigr)$.

Lemma 8.1.2 ÁöµM²³æ¦ý¬Û·í¹ê¥Î, ¥¦§i¶D§Ú­Ì¤¸¯À¶¡ªº¾ã°£Ãö«Y¥i¥HÂà´«¦¨ ideal ¶¡ªº¥]§tÃö«Y. ¥H«á§Ú­Ì­n½Í½×¨â¤¸¯À¶¡ªº¾ã°£Ãö«Y®É§Ú­Ì¦³®É¤£¥Î divisor ªº©w¸q³B²z, §Ú­Ì·|¥Î³oºØ ideal ªºÃö«Y¨Ó±´°Q, ¤j®a·|µo²³o­Ó¤èªk¬O²¼ä¤S¤è«Kªº.

­Y a $ \in$ R ¥B a$ \ne$ 0, §Ú­Ì«Ü§Öªº´Nª¾¹D¥ô·N R ¤¤ªº¤@­Ó unit ³£·|¬O a ªº¤@­Ó divisor. ³o¬O¥Ñ©ó­Y u ¬O R ¤¤ªº unit, «h $ \bigl($u$ \bigr)$ = R (Lemma 6.2.4). ¬G¥Ñ $ \bigl($a$ \bigr)$ $ \subseteq$ R = $ \bigl($u$ \bigr)$ ª¾ u | a. ¥t¤@¤è­±·í u ¬O unit ®É, a . u ¤]¬O a ªº divisor. ³o¤]¥i¥Ñ $ \bigl($a . u$ \bigr)$ = $ \bigl($a$ \bigr)$ (Lemma 6.5.4) ¤Î Lemma 8.1.2 °¨¤W±o¨ì. u ©M a . u ³oºØ a ªº divisor ¹ï a ªº¤À¸Ñ¨S¦³¬Æ»òÀ°§U, §Ú­ÌºÙ¤§¬° a ªº trivial divisor. ¥H¤U Lemma ¬O±´°Q a . u ³o­Ó a ªº trivial divisor ©M a ªºÂ²³æÃö«Y.

Lemma 8.1.3   ¥O R ¬O¤@­Ó integral domain ¥B a ©M b ¬O R ¤¤¨â­Ó¤£¬° 0 ªº¤¸¯À. ¤U¦C¤T¶µ a ©M b ªºÃö«Y¬Oµ¥»ùªº.
  1. ¦s¦b u $ \in$ R ¬O R ªº¤@­Ó unit º¡¨¬ a = b . u.
  2. $ \bigl($a$ \bigr)$ = $ \bigl($b$ \bigr)$.
  3. a | b ¥B b | a.

µý ©ú. (1) $ \Rightarrow$ (2): ¥i¥Ñ Lemma 6.5.4 ª¾ $ \bigl($a$ \bigr)$ = $ \bigl($b$ \bigr)$.

(2) $ \Rightarrow$ (3): ¥i¥Ñ Lemma 8.1.2 ª½±µ±À±o.

(3) $ \Rightarrow$ (1): ¥Ñ a | b ª¾¦s¦b r $ \in$ R ¨Ï±o b = a . r, ¦A¥Ñ b | a ª¾¦s¦b r' $ \in$ R ¨Ï±o a = b . r'. ¬Gª¾

a = b . r' = (a . r) . r' = a . (r . r').

¤]´N¬O»¡

a . (1 - r . r') = a - a . (r . r') = 0.

§Q¥Î a$ \ne$ 0 ¥B R ¬O¤@­Ó integral domain, ±o r . r' = 1. ´«¥y¸Ü»¡ r' ¬O R ªº¤@­Ó unit. $ \qedsymbol$

¬°¤F¤è«K°_¨£, §Ú­Ìµ¹¦³ Lemma 8.1.3 ¤¤ªºÃö«Y¤@­Ó¯S®íªº¦WºÙ.

Definition 8.1.4   ­Y a, b $ \in$ R $ \setminus$ {0} ¥B¦s¦b u $ \in$ R ¬O R ¤¤ªº¤@­Ó unit º¡¨¬ a = b . u, «hºÙ a ©M b ¬O associates. °O¬° a $ \sim$ b.

§Q¥Î Lemma 8.1.3 ¤¤ªº (2) §Ú­Ìª¾ a $ \sim$ b ­Y¥B°ß­Y $ \bigl($a$ \bigr)$ = $ \bigl($b$ \bigr)$, ©Ò¥H°¨¤W±oª¾ $ \sim$ ¬O¤@­Ó equivalence relation.

¦^ÅU¤@¤U¦b $ \mathbb {Z}$ ¤¤§Ú­Ì©w a, b ªº greatest common divisor ¬O a, b ªº common divisor ¤¤³Ì¤jªº, ¦Ó¦b F[x] ¤¤§Ú­Ì©w f (x), g(x) ªº greatest common divisor ¬O f (x), g(x) ªº common divisor ¤¤ degree ³Ì¤jªº. ¦b¤@¯ëªº integral domain ¬OµLªk©w¤j¤p©Î degree ªº. ¤£¹L«e¨âºØ±¡ªpªº greatest common divisor ³£¦³¤@­Ó¦@¦Pªº©Ê½è (°Ñ¨£ Corollary 7.1.5 (2) ¥H¤Î Corollary 7.2.9 (2)), §Ú­Ì´N¥Î³o­Ó©Ê½è¨Ó©w integral domain ¤¤ªº greatest common divisor.

Definition 8.1.5   ­Y R ¬O¤@­Ó integral domain, a1,..., an ¬O R ¤¤ªº«D 0 ¤¸¯À.
  1. ­Y c $ \in$ R º¡¨¬ c | ai,  $ \forall$ i $ \in$ {1,..., n} «hºÙ c ¬O a1,..., an ªº¤@­Ó common divisor.
  2. ­Y d $ \in$ R ¬O a1,..., an ªº¤@­Ó common divisor ¥Bº¡¨¬¹ï¥ô·N a1,..., an ªº common divisor c ¬Òº¡¨¬ c | d, «hºÙ d ¬O a1,..., an ªº¤@­Ó greatest common divisor.

­Y u ¬O R ¤¤ªº unit, «h¥Ñ©ó $ \bigl($u$ \bigr)$ = R (Lemma 6.2.4) ¥iª¾¹ï¥ô·N a1,..., an ¬Ò¦³ $ \bigl($ai$ \bigr)$ $ \subseteq$ $ \bigl($u$ \bigr)$, $ \forall$ i $ \in$ {1,..., n}. ¤]´N¬O»¡ u | ai, $ \forall$ i $ \in$ {1,..., n}. ¬Gª¾ R ¤¤ªº unit ³£¬O a1,..., an ªº common divisor. ¤£¹L¹ï¤@¯ëªº integral domain, ¹ï¥ô·Nªº a1,..., an ¨ä greatest common divisor ¥¼¥²¦s¦b. §Y¨Ï¦s¦b¨ä greatest common divisor ¤]¤£¤@©w°ß¤@ (¦b F[x] ªº±¡ªp´N¬O¤@¨Ò). ¥t¥ ­nª`·Nªº¬O¦b¦¹©w¸q¤§¤U $ \mathbb {Z}$ ¤¤ªº greatest common divisor ©M Section 7.1 ¤¤ Definition 7.1.3 ªº greatest common divisor ¬Û®t¤F¤@­Ó¥¿­t¸¹. ±µµÛ§Ú­Ì¦C¥X greatest common divisor ªº°ò¥»©Ê½è.

Lemma 8.1.6   ³] R ¬O¤@­Ó integral domain.
  1. °²³] d ©M d' ¬Ò¬° a1,..., an ªº greatest common divisor, «h d ©M d' associates.
  2. °²³] R ¤¤¥ô¨â­Ó«D 0 ¤¸¯Àªº greatest common divisor ¦s¦b, «h R ¤¤¥ô·N n ­Ó«D 0 ¤¸¯Àªº greatest common divisor ¤]¦s¦b.

µý ©ú. (1) ­Y d ©M d' ¬Ò¬O a1,..., an ªº greatest common divisor, «h¥Ñ©w¸qª¾ d ¬O a1,..., an ªº common divisor. ¦A§Q¥Î d' ¬O a1,..., an ªº greatest common divisor ±oÃÒ d | d'. ¦P²z±o d' | d. ¬G§Q¥Î Lemma 8.1.3 ª¾ d $ \sim$ d'.

(2) °²³] R ¤¤¥ô¨â­Ó«D 0 ¤¸¯Àªº greatest common divisor ¦s¦b, §Ú­Ì§Q¥Î¼Æ¾ÇÂk¯ÇªkÃÒ©ú¥ô·N n ­Ó«D 0 ¤¸¯À a1,..., an ªº greatest common divisor ¤]¦s¦b. °²³]¥ô·N n - 1 ­Ó«D 0 ¤¸¯À a1,..., an - 1 ªº greatest common divisor ¦s¦b¥B¬° d0. ¦] d0 ©M an ¬Ò¬O R ¤¤ªº«D 0 ¤¸¯À, ¥Ñ°²³]ª¾¨ä greatest common divisor ¦s¦b. ¥O d ¬° d0 ©M an ªº greatest common divisor, §Ú­Ì­nÃÒ©ú d ¬° a1,..., an ªº greatest common divisor.

­º¥ý¥Ñ d | d0 ¥B d0 ¬O a1,..., an1 ªº common divisor ª¾ d | d0 | ai, $ \forall$ i $ \in$ {1,..., n - 1}. ¦A¥Ñ d | an ª¾ d ¬O a1,..., an ªº¤@­Ó common divisor.

±µµÛ­Y c ¬O a1,..., an ªº¤@­Ó common divisor, «h c ·íµM¬O a1,..., an - 1 ªº¤@­Ó common divisor. ¬G¥Ñ d0 ¬O a1,..., an - 1 ªº greatest common divisor ª¾ c | d0. ´«¨¥¤§ c ¬O d0 ©M an ªº¤@­Ó common divisor. ¬G¥Ñ d ¬O d0 ©M an ªº greatest common divisor ª¾ c | d. ¦]¦¹¥Ñ©w¸qª¾ d ¬O a1,..., an ªº greatest common divisor. $ \qedsymbol$

³Ì«á§Ú­Ì­n©w¸q irreducible element ©M prime element. Irreducible ¬O¤£¥i¤À¸Ñªº·N«ä, ´«¨¥¤§´N¬O°£¤F trivial divisor ¥ ¨S¦³¨ä¥Lªº divisor.

Definition 8.1.7   ³] R ¬O¤@­Ó integral domain.
  1. ­Y a ¬O R ¤¤ªº«D 0 ¤¸¯À¥Bº¡¨¬ a ªº divisor ³£¬O trivial divisor (¤]´N¬O»¡, ­Y d | a «h d ¬O¤@­Ó unit ©Î d $ \sim$ a), «hºÙ a ¬O R ªº¤@­Ó irreducible element.
  2. ­Y p ¬O R ¤¤ªº«D 0 ¤¸¯À¥B¹ï¥ô·Nº¡¨¬ p | c . d ªº c, d $ \in$ R ¬Ò¦³ p | c ©Î p | d, «hºÙ p ¬O R ªº¤@­Ó prime element.

§Ú­Ì´£¹L irreducible element ©M prime element ªº©w¸q°ò¥»¤W¬O¤£¦Pªº, ©Ò¥H¥¦­Ì­ì«h¤W¬O¨âºØ¤£¦Pªº¯S©Ê. ¤£¹L¥H¤Uªºµ²ªG§i¶D§Ú­Ì¦b integral domain ¤§¤U prime element ¤@©w¬O irreducible element.

Lemma 8.1.8   °²³] R ¬O integral domain. ­Y a $ \in$ R ¬O¤@­Ó prime element, «h a ¤]¬O¤@­Ó irreducible element.

µý ©ú. ¥ô¨ú d | a, ­n»¡ a ¬O irreducible ´N¬O­nÃÒ©ú d ¬O¤@­Ó unit ©Î d $ \sim$ a. ¥Ñ©ó d | a, ¬G¦s¦b r $ \in$ R º¡¨¬ a = d . r. ©Ò¥H§Ú­Ì¦³ a | d . r. §Q¥Î a ¬O prime ªº©Ê½èª¾ a | d ©Î a | r. ¦pªG a | d, ¥Ñ d | a ªº°²³]¥H¤Î Lemma 8.1.3 ª¾ d $ \sim$ a. ¦pªG a | r, ¦P¼Ëªº¥Ñ Lemma 8.1.3 ª¾ a $ \sim$ r. ´«¥y¸Ü»¡, ¦s¦b¤@­Ó unit u ¨Ï±o a = u . r. ¥Ñ a = d . r = u . r ¥H¤Î R ¬O¤@­Ó integral domain ª¾ d = u ¬O¤@­Ó unit. $ \qedsymbol$

«e­±´¿¸g´£¹L§Ú­Ì³ßÅw¥Î ideal ªºÃö«Y¨Ó´yø¤¸¯À¶¡ªº¾ã°£Ãö«Y. ¤U­±ªº Lemma ´N¬O§i¶D§Ú­Ì irreducible element ©M prime element ©Ò²£¥Íªº principle ideal ©Ò¹ïÀ³ªº©Ê½è.

Lemma 8.1.9   °²³] R ¬O¤@­Ó integral domain, a $ \in$ R ¥B a$ \ne$ 0.
  1. a ¬O¤@­Ó irreducible element ­Y¥B°ß­Y¨S¦³ nontrivial principle ideal ¥]§t $ \bigl($a$ \bigr)$.
  2. a ¬O¤@­Ó prime element ­Y¥B°ß­Y $ \bigl($a$ \bigr)$ ¬O¤@­Ó prime ideal.

µý ©ú. (1) $ \Rightarrow$: °²³] a ¬O¤@­Ó irreducible element, ¦pªG¦s¦b b $ \in$ R º¡¨¬ $ \bigl($a$ \bigr)$ $ \subseteq$ $ \bigl($b$ \bigr)$, ¥Ñ Lemma 8.1.2 ª¾ b | a. ¬G¥Ñ a ¬O irreducible ±o b ¬O¤@­Ó unit ©Î¬O b $ \sim$ a. ´«¨¥¤§ $ \bigl($b$ \bigr)$ = R (Lemma 6.2.4) ©Î $ \bigl($b$ \bigr)$ = $ \bigl($a$ \bigr)$ (Lemma 6.5.4). ©Ò¥H§ä¤£¨ì nontrivial principle ideal ¥]§t $ \bigl($a$ \bigr)$.

$ \Leftarrow$: ¤Ï¤§­Y d | a, «hª¾ $ \bigl($a$ \bigr)$ $ \subseteq$ $ \bigl($d$ \bigr)$. ¥Ñ°²³]¨S¦³ nontrivial principle ideal ¥]§t $ \bigl($a$ \bigr)$, ±o $ \bigl($d$ \bigr)$ ¬O¤@­Ó trivial principle ideal ¥]§t $ \bigl($a$ \bigr)$. ´«¨¥¤§ $ \bigl($d$ \bigr)$ = R ©Î $ \bigl($d$ \bigr)$ = $ \bigl($a$ \bigr)$. ­Y $ \bigl($d$ \bigr)$ = R ªí¥Ü $ \bigl($d$ \bigr)$ = $ \bigl($1$ \bigr)$ ¬G¥Ñ Lemma 8.1.3 ª¾ d $ \sim$ 1, ¤]´N¬O»¡ d ¬O¤@­Ó unit. ­Y $ \bigl($d$ \bigr)$ = $ \bigl($a$ \bigr)$ ¦P¼Ë¥Ñ Lemma 8.1.3 ª¾ d $ \sim$ a. ¬G±o a ¬O¤@­Ó irreducible element.

(2) $ \Rightarrow$: °²³] a ¬O¤@­Ó prime element. ¦pªG c . d $ \in$ $ \bigl($a$ \bigr)$, ª¾ a | c . d. ¬G¥Ñ a ¬O prime ªº°²³]ª¾ a | c ©Î a | d. ³o§i¶D§Ú­Ì c $ \in$ $ \bigl($a$ \bigr)$ ©Î d $ \in$ $ \bigl($a$ \bigr)$, ¬G±oÃÒ $ \bigl($a$ \bigr)$ ¬O¤@­Ó prime ideal.

$ \Leftarrow$: °²³] $ \bigl($a$ \bigr)$ ¬O¤@­Ó prime ideal. ¥ô¨ú c, d $ \in$ R º¡¨¬ a | c . d, ª¾ c . d $ \in$ $ \bigl($a$ \bigr)$. ¬G¥Ñ $ \bigl($a$ \bigr)$ ¬O¤@­Ó prime ideal ªº°²³]±o c $ \in$ $ \bigl($a$ \bigr)$ ©Î d $ \in$ $ \bigl($a$ \bigr)$. ´«¨¥¤§ a | c ©Î a | d, ¬G±oÃÒ a ¬O¤@­Ó prime element. $ \qedsymbol$


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