By John E. Maxfield
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Extra resources for Abstract Algebra and Solution by Radicals
Why do we assume a ^ 0 in ax2 + bx + c = 0? Classroom Exercise 2-40. Learn prototypes for the abstract algebras you have studied, a familiar sample of each one. CHAPTER 3 MORE ABOUT GROUPS In the previous chapter we learned how groups fit into the picture of abstract algebras generally. They play such an important role that most of the algebras we defined are groups with further requirements. In this chapter we shall develop a way to take a group apart and look at its inner structure. Examples have shown us how a group can be represented by its “multi plication” table [more generally, its “operation” table, but we often use the familiar symbols of multiplication whatever the group operation is: a-1 for the inverse, ab for a 0 b, and 1 for the group identity].
This gives us a mapping from cities to temperatures. On a given day not all possible temperatures would be likely to appear, so the mapping would be into, not onto, the possible temperatures. Ordinarily there would not be an inverse mapping from temperatures to cities, as several cities would probably 28 O T H E R A B S T R A C T A LG EB R A S have the same temperature. (2) A listing of batting averages for baseball players gives a similar “into” mapping, ordinarily without an inverse mapping. v into the real numbers.
Theorem 2-1. If k , w, and n are in A, then (///) k(m + n) = km + kn. Proof by induction on k: Does (///) hold when k takes on the special value 1 ? We form k(m + n) at the value k = 1: Basis. k(m + n) |fc=1 = 1(m + n) = m + n, by (/) in the definition of multiplication, and km + kn |fc=1 = \m + In = m + n9 by two applications of (/). Comparing, we find that (///) holds for k = 1. O T H E R A BSTRA CT A LG EBR A S 23 Now we build up a subset S of N made up of all those values of k that make (iii) hold.
Abstract Algebra and Solution by Radicals by John E. Maxfield