[代數] 一題多項式想請教

看板Math作者 (瑋仔)時間14年前 (2011/05/12 22:34), 編輯推噓7(7推 0噓 24→)
留言31則, 3人參與, 最新討論串1/1
1.Prove: Let f(x),g(x) 屬於 F[x] Then gcd(f(x),g(x))=1 <＝﹥there exist a(x),b(x)屬於 F[x] such taht a(x)f(x)+b(x)g(x)=1 2.d(x)=a(x)f(x)+b(x)g(x) for some a(x),b(x)屬於 F[x] => d(x)=gcd(f(x),g(x)) ? -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 111.253.40.155
推
05/12 22:37, , 1F
2可能要加上deg(d(x))是最小的才對
05/12 22:37, 1F
推
05/12 22:43, , 2F
第一題能用這個定理嗎： f(x),g(x) 屬於 F[x]
05/12 22:43, 2F
→
05/12 22:43, , 3F
there exist a(x),b(x)屬於 F[x] s.t.
05/12 22:43, 3F
→
05/12 22:44, , 4F
d(x)=a(x)f(x)+b(x)g(x) , d(x)=gcd(f(x),g(x))
05/12 22:44, 4F
→
05/12 22:44, , 5F
如果就是要證這個定理 去翻書吧
05/12 22:44, 5F
→
05/12 22:50, , 6F
謝謝指教!!第一題基本上沒什麼問題
05/12 22:50, 6F
→
05/12 22:51, , 7F
第二題"=>"如果要成立的話要加上deg
05/12 22:51, 7F
→
05/12 22:52, , 8F
對嗎?
05/12 22:52, 8F
推
05/12 22:54, , 9F
恩
05/12 22:54, 9F
→
05/12 22:54, , 10F
我個人是這樣認為拉
05/12 22:54, 10F
→
05/12 22:55, , 11F
恩~~我在想看看
05/12 22:55, 11F
推
05/12 22:59, , 12F
第二題感覺有點奇怪耶 前提的d是指公因式嗎？
05/12 22:59, 12F
→
05/12 23:00, , 13F
如果題目是:for some a(x),b(x)屬於 F[x]
05/12 23:00, 13F
→
05/12 23:00, , 14F
a(x)f(x)+b(x)g(x) is the common divisor of f,g
05/12 23:00, 14F
→
05/12 23:01, , 15F
deg(d(x)) is the min degree of all common divisor
05/12 23:01, 15F
→
05/12 23:02, , 16F
這樣這句話變成廢話= =" 可是如果題目是
05/12 23:02, 16F
→
05/12 23:02, , 17F
d(x)=a(x)f(x)+b(x)g(x) for some a(x),b(x)€F[x]
05/12 23:02, 17F
→
05/12 23:03, , 18F
then d(x) is the common divisor of f and g
05/12 23:03, 18F
→
05/12 23:03, , 19F
那這個敘述就是錯的 x+x^2=1*x+1*x^2 可是x+x^2
05/12 23:03, 19F
→
05/12 23:04, , 20F
不是x與x^2的公因式 不知道是我誤會還是怎樣??
05/12 23:04, 20F
推
05/12 23:07, , 21F
因為他能造出來的d(x)如果是degree是最小的,那他就是
05/12 23:07, 21F
→
05/12 23:07, , 22F
common divisor
05/12 23:07, 22F
→
05/12 23:09, , 23F
有點玄妙就是了= =
05/12 23:09, 23F
→
05/12 23:34, , 24F
我解釋一下,我想問的是如果d(x)可以表示成
05/12 23:34, 24F
→
05/12 23:35, , 25F
d(x)=a(x)f(x)+b(x)g(x) 那麼d(x)就可以表示f,g的
05/12 23:35, 25F
→
05/12 23:35, , 26F
最大公因式??
05/12 23:35, 26F
→
05/12 23:36, , 27F
我覺得是不行,可是我不知道缺了什麼條件.........
05/12 23:36, 27F
推
05/12 23:37, , 28F
x+x^2=1*x+1*x^2 , d=x+x^2 , f=x , g=x^2
05/12 23:37, 28F
→
05/12 23:38, , 29F
(f,g)=x 而d=x+x^2 卻也代表不了什麼東西
05/12 23:38, 29F
推
05/12 23:43, , 30F
要推到是最大公因式的話要加上那個d(x)的degree是線
05/12 23:43, 30F
→
05/12 23:43, , 31F
性組合中最小的應該就是最大公因式了
05/12 23:43, 31F
‣ 更多 james2009 的文章
文章代碼(AID): #1Do-_ciW (Math)