※ 引述《Leon (Achilles)》之銘言：
: 某日在上課時, 老師靈光一閃, 補充了這個題目.
: Problem :
: f is a polynomial in GF(2), which f = X^N + X^a + 1.
: Try to prove that f can not be devided by X^4 + X^3 + X^2 + X + 1.
這真是篇很久的文
現在才看到發現很有趣
f can be written as following:
x^N + x^a + 1 = (x^5 + 1)*p(x) + r(x)
= (x^4 + x^3 + x^2 + x + 1)*q(x) + r(x), --- (1)
where q(x) = (x + 1)*p(x)
as long as we show the remainder r(x) can't be divided by
(x^4 + x^3 + x^2 + x + 1) in (1), we then prove the polynomial f
can't be divided by (x^4 + x^3 + x^2 + x + 1).
since r(x) is the remainder of x^N + x^a + 1 when the divisor is (x^5 + 1)
, therefore, it's evident that the reaminder r(x) should be as the form:
r(x) = x^M + x^b + 1, where M,b < 5.
now, it's clear that r(x) in (1) can't be the multiple of
(x^4 + x^3 + x^2 + x + 1), thus we finish the proof.
還請Leon兄指教
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