By John E. Maxfield

ISBN-10: 0486671216

ISBN-13: 9780486671215

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**Extra info for Abstract Algebra and Solution by Radicals**

**Sample text**

See Definition 1-2 ii). One way of adjoining an additive identity to N is to lump all these problems together in an “equivalence class” and equate the answer to the problems: Dropping all but the pair of numbers (

F IG U R E 2-3 J 36 O T H E R A B ST R A C T A LG E B R A S between <2i and I. According to the operation -f defined on g , we have a 1 b __ a - 1 + 1 • b _ a + b 1“ 11 “ 1 Then, adding a and b within / according to the definition of + in /, we obtain a + b. The two sums then do correspond according to (1). The product as defined in Q is a . b _ a • b _ ab_ 1 1~ 1•1“ 1 ’ which corresponds according to (1) to the product ab as defined in /. This shows that the correspondence is an isomorphism and so justifies also the assertion that Qi is a ring, since all the ring properties are properties of + and *, which Qx has been shown to share with /, a known ring.

We let i stand for the pair (0, 1), noting that (0, l)2 = ( - 1 , 0). Theorem 2-7. The complex numbers with + and •, as defined in Definition 2-17, constitute a field. P roof : Directly from the definition we can see that addition and multipli cation are operations on C. Exercise 2-32. What properties of arithmetic in R do you need to prove the associative law of addition in C? Exercise 2-33. What properties of arithmetic in R do you need to prove the associative law of multiplication in C? O T H E R A B S T R A C T A LG E B R A S 43 Exercise 2-34.

### Abstract Algebra and Solution by Radicals by John E. Maxfield

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