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- is x^3 + 1 irreducible over Z_3?Suppose the multiplication cx is defined to produce (cx1, 0) instead of (cx1, cx2 ). With the usual addition in R 2 , are the eight conditions satisfied?Suppose that L and K are subfields of GF(pn). If L has ps elements and K has pt elements, how many elements does L ⋂ K have?
- Let q1, q2, q3, . . . be defined by q1 = 2, qn = 3qn−1 − 1 for n ≥ 2. Show byinduction that for all n ≥ 1, qn =1/2(3n + 1).5. For f(x) = 1 –x2on [−2, 1], do the hypotheses and conclusion of Rolle’s Theorem hold?b) If (an) has limit−1 and (bn) tends to infinity, does it follow that only finitely many terms of (an bn) are positive?