Let FFF be the field ({0,1,2},+,⋅)(\{0,1,2\}, +, \cdot)({0,1,2},+,⋅), where +++ is the mod-3 addition and ⋅\cdot⋅ is the mod-3 multiplication. If the polynomial
p(x)=a+bx+cx2∈P(F)p(x) = a + bx + cx^2 \in P(F)p(x)=a+bx+cx2∈P(F)
satisfies p(i)=i3+2p(i) = i^3 + 2p(i)=i3+2 for each i∈Fi \in Fi∈F, then (a,b,c)(a,b,c)(a,b,c) is ___.
參考答案與解析
(a,b,c)=(2,1,0)(a,b,c)=(2,1,0)(a,b,c)=(2,1,0)。在mod 3體F={0,1,2}F=\{0,1,2\}F={0,1,2}上代入三個條件：p(0)=a=03+2=2p(0)=a=0^3+2=2p(0)=a=03+2=2；p(1)=a+b+c=13+2≡0(mod3)p(1)=a+b+c=1^3+2\equiv0\pmod3p(1)=a+b+c=13+2≡0(mod3)；p(2)=a+2b+4c≡23+2=10≡1(mod3)p(2)=a+2b+4c\equiv2^3+2=10\equiv1\pmod3p(2)=a+2b+4c≡23+2=10≡1(mod3)。解此三元一次聯立方程（mod 3）得a=2,b=1,c=0a=2,b=1,c=0a=2,b=1,c=0，即p(x)=2+xp(x)=2+xp(x)=2+x（已逐一代入i=0,1,2i=0,1,2i=0,1,2驗證p(i)≡i3+2(mod3)p(i)\equiv i^3+2\pmod3p(i)≡i3+2(mod3)成立）。
