º¥ý§ÚÌÁÙ¬O¹ï¤@Ó¤¸¯Àªº¦]¼Æµ¹¤@Ó¥¿¦¡ªº©w¸q.
¦^ÅU¤@¤UY R ¬O integral domain ¥B d R, «h d = {d . r | r R} ©Ò¥H¥Ñ¤W¤@Ó©w¸q§ÚÌ«Ü®e©öª¾ d | a Y¥B°ßY a d. µM¦ÓY a d, ¥Ñ d ¬O¤@Ó ideal ª¾¹ï¥ô·Nªº r R ¬Ò¦³ a . r d. ¬G±o a d. ¤Ï¤§Y a d, ¥Ñ a a ±oª¾ a d. ´«¥y¸Ü»¡ a d Y¥B°ßY a d, ¦]¦¹§Ú̦³¥H¤Uªºµ²½×:
Lemma 8.1.2 ÁöµM²³æ¦ý¬Û·í¹ê¥Î, ¥¦§i¶D§Ṳ́¸¯À¶¡ªº¾ã°£Ãö«Y¥i¥HÂà´«¦¨ ideal ¶¡ªº¥]§tÃö«Y. ¥H«á§ÚÌn½Í½×¨â¤¸¯À¶¡ªº¾ã°£Ãö«Y®É§Ú̦³®É¤£¥Î divisor ªº©w¸q³B²z, §ÚÌ·|¥Î³oºØ ideal ªºÃö«Y¨Ó±´°Q, ¤j®a·|µo²³oÓ¤èªk¬O²¼ä¤S¤è«Kªº.
Y a R ¥B a 0, §Ú̫ܧ֪º´Nª¾¹D¥ô·N R ¤¤ªº¤@Ó unit ³£·|¬O a ªº¤@Ó divisor. ³o¬O¥Ñ©óY u ¬O R ¤¤ªº unit, «h u = R (Lemma 6.2.4). ¬G¥Ñ a R = u ª¾ u | a. ¥t¤@¤è±·í u ¬O unit ®É, a . u ¤]¬O a ªº divisor. ³o¤]¥i¥Ñ a . u = a (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, §Ú̺٤§¬° a ªº trivial divisor. ¥H¤U Lemma ¬O±´°Q a . u ³oÓ a ªº trivial divisor ©M a ªºÂ²³æÃö«Y.
(2) (3): ¥i¥Ñ Lemma 8.1.2 ª½±µ±À±o.
(3) (1): ¥Ñ a | b ª¾¦s¦b r R ¨Ï±o b = a . r, ¦A¥Ñ b | a ª¾¦s¦b r' R ¨Ï±o a = b . r'. ¬Gª¾
¬°¤F¤è«K°_¨£, §Ú̵¹¦³ Lemma 8.1.3 ¤¤ªºÃö«Y¤@Ó¯S®íªº¦WºÙ.
§Q¥Î Lemma 8.1.3 ¤¤ªº (2) §Ú̪¾ a b Y¥B°ßY a = b, ©Ò¥H°¨¤W±oª¾ ¬O¤@Ó equivalence relation.
¦^ÅU¤@¤U¦b ¤¤§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¥Î³oөʽè¨Ó©w integral domain ¤¤ªº greatest common divisor.
Y u ¬O R ¤¤ªº unit, «h¥Ñ©ó u = R (Lemma 6.2.4) ¥iª¾¹ï¥ô·N a1,..., an ¬Ò¦³ ai u, i {1,..., n}. ¤]´N¬O»¡ u | ai, i {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 ¤¤ªº greatest common divisor ©M Section 7.1 ¤¤ Definition 7.1.3 ªº greatest common divisor ¬Û®t¤F¤@Ó¥¿t¸¹. ±µµÛ§Ú̦C¥X greatest common divisor ªº°ò¥»©Ê½è.
(2) °²³] R ¤¤¥ô¨âÓ«D 0 ¤¸¯Àªº greatest common divisor ¦s¦b, §Ú̧Q¥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, i {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.
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.
§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§Ú̦b integral domain ¤§¤U prime element ¤@©w¬O irreducible element.
«e±´¿¸g´£¹L§Ú̳ßÅw¥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.
: ¤Ï¤§Y d | a, «hª¾ a d. ¥Ñ°²³]¨S¦³ nontrivial principle ideal ¥]§t a, ±o d ¬O¤@Ó trivial principle ideal ¥]§t a. ´«¨¥¤§ d = R ©Î d = a. Y d = R ªí¥Ü d = 1 ¬G¥Ñ Lemma 8.1.3 ª¾ d 1, ¤]´N¬O»¡ d ¬O¤@Ó unit. Y d = a ¦P¼Ë¥Ñ Lemma 8.1.3 ª¾ d a. ¬G±o a ¬O¤@Ó irreducible element.
(2) : °²³] a ¬O¤@Ó prime element. ¦pªG c . d a, ª¾ a | c . d. ¬G¥Ñ a ¬O prime ªº°²³]ª¾ a | c ©Î a | d. ³o§i¶D§ÚÌ c a ©Î d a, ¬G±oÃÒ a ¬O¤@Ó prime ideal.
: °²³] a ¬O¤@Ó prime ideal. ¥ô¨ú c, d R º¡¨¬ a | c . d, ª¾ c . d a. ¬G¥Ñ a ¬O¤@Ó prime ideal ªº°²³]±o c a ©Î d a. ´«¨¥¤§ a | c ©Î a | d, ¬G±oÃÒ a ¬O¤@Ó prime element.