[功課] Irreducible polynomial in finite field
3.8最後一題習題的最後一小提
Prove, for all n≧1 and finite field k,
there is an irreducible polynomial in k[x]
Proof:
Let F_p be the prime of k, and so that |k|=p^r, for some r>0.
Let F_p^{nr} be the field having p^{nr} elements.
By thm, for all z in F_p^{nr},z is a root fo f(x)=x^(p^{nr})-x
and since p^r-1 | p^{nr}-1, so k is a subfield of F_p^{nr}
and [F_p^{nr},k][k,F_p]=[F_p^{nr},F_p]
[F_p^{nr},k]*r=nr
[F_p^{nr},k]=n
Therefore, there exists a irreducible polynomial g(x) of degree n
in k[x] such that F_p^{nr} has a root of g(x)
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◆ From: 123.193.85.88
※ 編輯: jacky7987 來自: 123.193.85.88 (05/29 01:39)
※ 編輯: jacky7987 來自: 123.193.85.88 (05/29 01:40)
推
05/29 10:50, , 1F
05/29 10:50, 1F
推
05/29 15:38, , 2F
05/29 15:38, 2F
推
05/29 22:17, , 3F
05/29 22:17, 3F
※ 編輯: jacky7987 來自: 123.193.89.201 (05/29 22:46)
→
05/29 22:47, , 4F
05/29 22:47, 4F
→
05/29 22:47, , 5F
05/29 22:47, 5F
→
05/29 22:48, , 6F
05/29 22:48, 6F
→
05/29 22:48, , 7F
05/29 22:48, 7F
→
05/29 22:49, , 8F
05/29 22:49, 8F
→
05/29 22:49, , 9F
05/29 22:49, 9F
→
05/29 22:50, , 10F
05/29 22:50, 10F
推
05/29 23:07, , 11F
05/29 23:07, 11F
→
05/29 23:09, , 12F
05/29 23:09, 12F
→
05/29 23:10, , 13F
05/29 23:10, 13F
→
05/29 23:10, , 14F
05/29 23:10, 14F
→
05/29 23:11, , 15F
05/29 23:11, 15F
→
05/29 23:12, , 16F
05/29 23:12, 16F
推
05/29 23:16, , 17F
05/29 23:16, 17F
→
05/29 23:17, , 18F
05/29 23:17, 18F
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