# Algebra (Prindle, Weber and Schmidt Series in Advanced by Mark Steinberger

By Mark Steinberger

The purpose of this booklet is to introduce readers to algebra from some degree of view that stresses examples and type. at any time when attainable, the most theorems are taken care of as instruments that could be used to build and examine particular sorts of teams, jewelry, fields, modules, and so forth. pattern buildings and classifications are given in either textual content and workouts.

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**Example text**

But then p1 is divisible by q1 , and since both are prime, they must be equal. We then have p1 (1 − q1s1 −1 . . qlsl ) = 0, so that (1 − q1s1 −1 . . qlsl ) = 0, and hence q1s1 −1 . . qlsl = 1. Since the qi are prime and since no positive number divides 1 other than 1 itself, we must have l = 1 and s1 = 1. Suppose t > 1. 16, together with an induction on s1 + · · · + sl , shows that p1 must divide qi for some i. But since p1 and qi are prime, we must have p1 = qi . A similar argument shows that q1 = pj for some j, and our order assumption then shows that i = j = 1, so that p1 = q1 .

Proof (a + b) + c = a + b + c = a + b + c. Since a + (b + c) has the same expansion, addition is associative. The other assertions follow by similar arguments. 7 says, simply, that the operations of addition and multiplication give Zn the structure of a commutative ring. When we refer to Zn as a group, we, of course, mean with respect to addition. 4, Zn has order n. 8. Let n be a positive integer. Then there are groups of order n. 4, Zn = Z/ ≡, where ≡ denotes equivalence modulo n. CHAPTER 2. 9.

3 shows that Rθ ·Rφ = Rφ ·Rθ . Thus, unlike Gl2 (R), SO(2) is an abelian group. 6. There is a homomorphism, exp, from R onto SO(2), deﬁned by exp(θ) = Rθ . The kernel of exp is 2π = {2πk | k ∈ Z}. Thus, Rθ = Rφ if and only if θ − φ = 2πk for some k ∈ Z. 3. Suppose that θ ∈ ker exp. Since the identity element of SO(2) is the identity matrix, the deﬁnition of Rθ forces cos θ to be 1 and sin θ to be 0. 9, we have that exp θ = exp φ if and only if −φ + θ ∈ ker(exp), so the result follows. We conclude with a calculation of the order of the elements of SO(2).