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In trouble-free introductions to mathematical research, the remedy of the logical and algebraic foundations of the topic is unavoidably really skeletal. This ebook makes an attempt to flesh out the bones of such therapy by way of offering a casual yet systematic account of the rules of mathematical research written at an straightforward point.

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Show that f = p 1 f 1 + p2 f 2 + · · · + pk f k . 9. Let f ∈ F [x] have derivative f . Then f is a product of distinct irreducible polynomials over F if and only if f and f are relatively prime. 10. Euclidean algorithm for polynomials Let f (x) and g(x) be polynomials over F for which deg(f (x)) ≥ deg(g(x)) ≥ 1. Use the division algorithm for polynomials to compute polynomials qi (x) and ri (x) as follows. f (x) g(x) r1 (x) r2 (x) = = = = .. rj (x) = rj (x) = q1 (x)g(x) + r1 (x), degr1 (x) < deg(g(x)).

Now we turn our attention to the construction of an algebra which is quite different from the two just given. Let F be a field and let S be the set of all nonnegative integers. We have seen that the set of all functions from S into F is a vector space which we now denote by F ∞ . , lists) f = (f0 , f1 , f2 , . ) of scalars fi ∈ F . If g = (g0 , g1 , g2 , . ) and a, b ∈ F , then af + bg is the infinite list given by af + bg = (af0 + bg0 , af1 + bg1 , . 1) We define a product in F ∞ by associating with each pair (f, g) of vectors in F ∞ the vector f g which is given by n (f g)n = fi gn−i , n = 0, 1, 2, .

Let u be any vector of U , so u = ni=1 ai ui for unique scalars ai ∈ F . Then the desired T has to be defined by T (u) = ni=1 ai T (ui ) = ni=1 ai vi . This clearly defines T uniquely. The fact that T is linear follows easily from the basic properties of vector spaces. Put L(U, V ) = {T : U → V : T is linear}. The interesting fact here is that L(U, V ) is again a vector space in its own right. Vector addition S + T is defined for S, T ∈ L(U, V ) by: (S + T )(u) = S(u) + T (u) for all u ∈ U . Scalar multiplication is defined for a ∈ F and T ∈ L(U, V ) by (aT )(u) = a(T (u)) for all u ∈ U .