Re: [代數] a*1+b*2^1/3+c*2^2/3最小多項式係數
※ 引述《ma4wanderer (師大之狼)》之銘言：
: 大家好
: K=A+B*2^(1/3)+C*2^(2/3)，
: A B C是有理數
: 想求X在Q中的最小多項式
: 求其他兩個根很簡單，可是如果做
: (X-K1)(X-K2)(X-K3) 計算有夠累
: 計算出x^2項的係數應該是-3A，常數項是A^3+2B^3+4C^3
: 想問各位大大有沒有什麼更好的計算方式？
Firstly, it is easy to find that g(x)=x^3-2 is the minimal polynomial of 2^(1/3) over Q (rational number field), and denote by x_1 and x_2 the two complex roots of x^3-2 other than 2^(1/3).
Consider the field F=Q[2^(1/3)]. There are three embedings of F into C (complex number field) which fix Q, i.e., {1,f_1,f_2}, where 1 is the identity map, f_1 maps 2^(1/3) to x_1, and f_2 maps 2^(1/3) to x_2.
Now the minimal polynomial of K over Q is exactly (x-K)(x-f_1(K))(x-f_2(K))=(x-A-B*2^(1/3)-C*2^(2/3))(x-A-B*x_1-C*(x_1)^2)(x-A-B*x_2-C*(x_2)^2).
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推
02/18 00:25, , 1F
02/18 00:25, 1F
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02/18 00:36, , 2F
02/18 00:36, 2F
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02/18 00:39, , 3F
02/18 00:39, 3F
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02/18 00:43, , 4F
02/18 00:43, 4F
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10/21 21:01, , 5F
10/21 21:01, 5F
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10/21 21:02, , 6F
10/21 21:02, 6F
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10/21 21:03, , 7F
10/21 21:03, 7F
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10/22 02:38, , 8F
10/22 02:38, 8F
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10/22 02:48, , 9F
10/22 02:48, 9F
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10/22 06:26, , 10F
10/22 06:26, 10F
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10/22 14:07, , 11F
10/22 14:07, 11F
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10/22 16:48, , 12F
10/22 16:48, 12F
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10/22 18:46, , 13F
10/22 18:46, 13F
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10/22 19:32, , 14F
10/22 19:32, 14F
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10/22 19:33, , 15F
10/22 19:33, 15F
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10/22 19:35, , 16F
10/22 19:35, 16F
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10/22 19:49, , 17F
10/22 19:49, 17F
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10/22 19:51, , 18F
10/22 19:51, 18F
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10/22 19:52, , 19F
10/22 19:52, 19F
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10/22 19:55, , 20F
10/22 19:55, 20F
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10/22 19:55, , 21F
10/22 19:55, 21F
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10/22 19:56, , 22F
10/22 19:56, 22F
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10/22 20:04, , 23F
10/22 20:04, 23F
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10/22 20:06, , 24F
10/22 20:06, 24F
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10/22 20:07, , 25F
10/22 20:07, 25F
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10/22 20:08, , 26F
10/22 20:08, 26F
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10/22 20:41, , 27F
10/22 20:41, 27F
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10/22 21:51, , 28F
10/22 21:51, 28F
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10/22 23:12, , 29F
10/22 23:12, 29F
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10/23 17:25, , 30F
10/23 17:25, 30F
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10/23 17:27, , 31F
10/23 17:27, 31F
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10/23 17:29, , 32F
10/23 17:29, 32F
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10/23 17:32, , 33F
10/23 17:32, 33F
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10/23 18:03, , 34F
10/23 18:03, 34F
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10/23 18:27, , 35F
10/23 18:27, 35F
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10/23 20:50, , 36F
10/23 20:50, 36F
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10/23 20:51, , 37F
10/23 20:51, 37F
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10/23 22:20, , 38F
10/23 22:20, 38F
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10/23 22:24, , 39F
10/23 22:24, 39F
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10/24 14:29, , 40F
10/24 14:29, 40F
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10/24 14:59, , 41F
10/24 14:59, 41F
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10/24 15:17, , 42F
10/24 15:17, 42F
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10/24 16:10, , 43F
10/24 16:10, 43F
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10/24 19:57, , 44F
10/24 19:57, 44F
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10/24 21:46, , 45F
10/24 21:46, 45F
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10/24 22:11, , 46F
10/24 22:11, 46F
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10/24 23:22, , 47F
10/24 23:22, 47F
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10/24 23:23, , 48F
10/24 23:23, 48F
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10/24 23:32, , 49F
10/24 23:32, 49F
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10/25 00:13, , 50F
10/25 00:13, 50F
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10/25 00:38, , 51F
10/25 00:38, 51F
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10/25 00:38, , 52F
10/25 00:38, 52F
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10/25 00:42, , 53F
10/25 00:42, 53F
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10/25 01:19, , 54F
10/25 01:19, 54F
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10/25 02:00, , 55F
10/25 02:00, 55F
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10/25 12:06, , 56F
10/25 12:06, 56F
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10/25 12:23, , 57F
10/25 12:23, 57F
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10/25 13:40, , 58F
10/25 13:40, 58F
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10/25 13:40, , 59F
10/25 13:40, 59F
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10/25 20:47, , 60F
10/25 20:47, 60F
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10/26 12:53, , 61F
10/26 12:53, 61F
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10/26 13:41, , 62F
10/26 13:41, 62F
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10/26 14:40, , 63F
10/26 14:40, 63F
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10/26 16:54, , 64F
10/26 16:54, 64F
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10/26 22:32, , 65F
10/26 22:32, 65F
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10/26 22:34, , 66F
10/26 22:34, 66F
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10/26 22:35, , 67F
10/26 22:35, 67F
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10/26 22:56, , 68F
10/26 22:56, 68F
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10/26 22:56, , 69F
10/26 22:56, 69F
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10/26 23:00, , 70F
10/26 23:00, 70F
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10/26 23:01, , 71F
10/26 23:01, 71F
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10/27 12:27, , 72F
10/27 12:27, 72F
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10/27 17:28, , 73F
10/27 17:28, 73F
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10/27 21:59, , 74F
10/27 21:59, 74F
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10/27 23:25, , 75F
10/27 23:25, 75F
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10/28 19:08, , 76F
10/28 19:08, 76F
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