[代數] 有關代數的幾個問題
1. show that a group G is abelian iff x^2=e
for any x in g, where e is identity of G
這題我可以證明x^2=e -> ab=ba 但是反方向我就不知道怎麼寫了
且在實數的運算上 不是就會滿足交換率 但並沒有每個數字都滿足x^2=e
例如2*3=3*2 但是 2^2=/=3^2=/=e 不知道是我哪裡理解錯了
2. a) construct a field F over Q such that x^7+2x+2 has a root in F
find the degree of extension of F over Q
b) construct a finite field of 27 elements
a)的部份我只知道by Eisenstein irreducibility criterion 可以看出
他在Q上沒有解, 且因代數基本定理知道他最少在C上一定有解
可是我要怎麼知道它是在R中有解還是C中有解?
b)的部份不確定F=Z3*Z3*Z3可以是一個答案嗎?
因為某些原因所以我代數這裡都要自學所以可能會問一些很蠢的問題
希望板友們可以包容
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◆ From: 1.34.20.166
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06/14 13:06, , 1F
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06/14 13:09, , 2F
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06/14 13:10, , 3F
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06/14 13:11, , 4F
06/14 13:11, 4F
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06/14 13:12, , 5F
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06/14 13:12, , 6F
06/14 13:12, 6F
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06/14 13:13, , 7F
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06/14 13:15, , 8F
06/14 13:15, 8F
感謝板友的指教 我去看了一下Kronecker's Theorem
因為這題他只有這樣然後是一大題 所以應該不能直接說by Kronecker's Theorem
就直接給出答案 我有參考了那個定理寫出答案
但不確定這樣可不可
let p(x)=x^7+2x+2 & it is irreducible over Q
by Eisenstein irreducibility criterion
and Q[x]/p(x) is a field iff p(x) is irreducible
set F = Q[x]/p(x) , and therefore F is a field
regard F is a extension of Q
_ ____ _
x =x+p(x) p(x)=0 in F
________ _ _ _ _
x^7+2x+2=0 => x^7+2x+2=0
_
hence x is a root of p(x) in F
element of F is of a0+a1x+...a6x^6 this form
so the degree of F is 7
※ 編輯: jas0205 來自: 1.34.20.166 (06/14 14:46)
※ 編輯: jas0205 來自: 1.34.20.166 (06/14 14:50)