By Dupre M.J., Glazebrook J.F., Prevlato E.
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Extra info for A Banach Algebra Version of the Sato Grassmannian and Commutative Rings of Differential Operators
In this chapter we do not prove F [x] is a unique factorization domain, nor do we even define unique factorization domain. The next definition and theorem are included merely for reference, and should not be studied at this stage. Definition Suppose F is a field and f ∈ F [x] has degree ≥ 1. The statement that g is an associate of f means ∃ a unit u ∈ F [x] such that g = uf . The statement that f is irreducible means that if h is a non-constant polynomial which divides f , then h is an associate of f .
Vn are non-negative integers, then ax1v1 x2v2 · · · xnvn is called a monomial. Order does not matter here. , xn ] to be any finite sum of monomials. , xn−1 ])[xn ]. Using this and induction on n, it is easy to prove the following theorem. Theorem units of R. , xn ] is a domain and its units are just the Chapter 3 Rings 49 Exercise Suppose R is a commutative ring and f : R[x, y] → R[x] is the evaluation map which sends y to 0. This means f (p(x, y)) = p(x, 0). Show f is a ring ¯ ¯ homomorphism whose kernel is the ideal (y) = yR[x, y].
Show f Then G is a group under addition. Define f : G → G by f (h) = dh dt is a homomorphism and find its kernel and image. Let g : [0, 1] → R be defined by g(t) = t3 − 3t + 4. Find f −1 (g) and show it is a coset of ker(f ). Exercise Let G be as above and g ∈ G. Define f : G → G by f (h) = h + 5h + 6t2 h. Then f is a group homomorphism and the differential equation h +5h +6t2 h = g has a solution iff g lies in the image of f . Now suppose this equation has a solution and S ⊂ G is the set of all solutions.