(2) Let F denote a field. Recall that P2(F) denotes the vector space of polynomi
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Question
(2) Let F denote a field. Recall that P2(F) denotes the vector space of polynomials in one variable having degree less than or equal to 2 and with coefficients in F; and F(F, F) denotes the vector space of all maps h: F --> F. Note that any polynomial ax2+bx+c in P2(F) may also be regarded as the vector h in F(F, F) defined by h(?)=a?2+b?+c for each ? in F.(b) If F denotes the integers mod 2, then show that there are two polynomials in P2(F) which are independent in P2(F) but are not independent in F(F, F).
Explanation / Answer
The polynomial f(T) = T 2 + T + 1 is irreducible over Z/2Z, and (Z/2Z)[T] / (T2+T+1) has size 4. Its elements can be written as the set {0, 1, t, t+1} where the multiplication is carried out by using the relation t2 + t + 1 = 0. In fact, since we are working over Z/2Z (that is, in characteristic 2), we may write this as t2 = t + 1. (This follows because -1 = 1 in Z/2Z) Then, for example, to determine t3, we calculate: t3 = t(t2) = t(t+1) = t2+t = t+1+t = 2t + 1 = 1, so t3 = 1.
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