Math 515 homework #1 Due date: 19980119 D. J. Bernstein 19980112 Let R be a set with elements 0 and 1, a unary operation − (“negation”), and binary operations + (“addition”) and · (“multiplication”). R is a ring if it satisfies the following identities: 0 + r = r; 1 · r = r; (−r) + r = 0; r + (s + t) = (r + s) + t; r · (s · t) = (r · s) · t; r · (s + t) = (r · s) + (r · t); r + s = s + r; and r · s = s · r. In particular, R is a commutative group under 0, −, +. For example, C, the set of complex numbers, is a ring. Notation: r · s is abbreviated rs, and r + (−s) is abbreviated r − s. Multiplication has higher precedence than addition; e.g., r · s + t = (r · s) + t.

Finite sums. Let X be a finite set, and let R be a commutative group under 0, −, +. Recall that any function f : X → R has a unique sum ∑_{x∈X} f(x) ∈ R. Here x is a dummy variable. Some useful identities: ∑_{x∈{a}} f(x) = f(a); ∑_{x∈X} f(x) + ∑_{x∈Y} f(x) = ∑_{x∈X∪Y} f(x) + ∑_{x∈X∩Y} f(x); ∑_{x∈X}(f(x)+g(x)) = ∑_{x∈X} f(x)+∑_{x∈X} g(x); ∑_{x∈X} f(x) = 0 if f(x) = 0 for all x ∈ X; and ∑_{x∈X} f(x) = ∑_{i∈I} ∑_{x∈S(i)} f(x) if I is a finite set and X is the disjoint union of the sets S(i) for i ∈ I. If R is a ring then r ∑_{x∈X} f(x) = ∑_{x∈X} rf(x) for any r ∈ R. Notation: If X = {x : p(x)} where p is a proposition, then ∑_{x∈X} is abbreviated ∑_{x:p(x)}, or ∑_{p(x)} when it is clear that x is the dummy variable. For example, ∑_{0≤i≤9} means ∑_{i∈{0,1,2,...,9}}.

Power series. Let R be a ring. A power series over R is an element of R^{N}; here N = {0, 1, 2, …}. In other words, a power series over R is a sequence (a_{0}, a_{1}, a_{2}, …) with each a_{i} ∈ R. Let f = (f_{0}, f_{1}, …) and g = (g_{0}, g_{1}, …) be power series over R. The sum of f and g, written f + g, is the power series (f_{0} + g_{0}, f_{1} + g_{1}, …). The product of f and g, written f · g, is the power series (p_{0}, p_{1}, …) with p_{n} = ∑_{j+k=n} f_{j}g_{k}. Exercise 1. Let f, g, h be power series over a ring R. Prove that (f · g) · h = f · (g · h). Consequently the set of power series over R, with addition and multiplication defined above, 0 = (0, 0, 0, …), 1 = (1, 0, 0, …), and −(a_{0}, a_{1}, …) = (−a_{0}, −a_{1}, …), is a ring. Notation: The ring of power series over R is written R[[x]], with a dummy variable x. The series (0, 1, 0, 0, …) is abbreviated x. Any series f = (f_{0}, f_{1}, …) is written ∑_{n≥0} f_{n}x^{n}. (There is a natural metric on R[[x]], to be discussed later, in which f_{0} + f_{1}x + f_{2}x^{2} + ⋯ converges to f.)

Multiples. Let R be a commutative group under 0, −, +. Let b be an integer. Recall that multiplication by b is a function R → R, written s ↦ bs, defined by bs = ∑_{i∈{1,2,...,b}} s if b ≥ 0 and bs = −(−b)s if b < 0. Some useful identities: 1s = s; (a + b)s = as + bs; (ab)s = a(bs); b(s + t) = bs + bt.

Derivatives. The derivative of a power series f = ∑_{n≥0} f_{n}x^{n} ∈ R[[x]] is f′ = ∑_{n≥0}(n+1)f_{n+1}x^{n} ∈ R[[x]]. Exercise 2. Prove that (fg)′ = f′g + g′f for f, g ∈ R[[x]].