Cohn P.M.'s Algebraic Numbers and Algebraic Functions PDF

By Cohn P.M.

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0 = lab. b) For a ;o! 0, prove that I"" ill both one-to-one and onto. c) For a ;o! 0, obtain 1.. 1• 9. Using fllnl'tions/: R' -- R', give an example of a function which ill a) one-to-one but not onto, b) onto but not one-to-one. 10. tioll, then I ill 8\1111. b) If log i,.. ion, th(\n g ill aillo. Iwn log iH I~IHCI nnu-tn-one and (jo g)-I = g-I 0/-1. + 0 0 (/ 11. Establish the following characterizations for any function I: x. -- Y: ill onto Y if and only if for all functions g, h: Y -+ Z, g. I = hoi implies that g = h.

Conversely, such a table could equally well serve to define a binary operation on 8, for the re8ult, of ('ombining any pair of elements of S would be displayed somewhere in tllP tahle. Example 2-2. he column marked 3. son to expect that a • b will be the Rllllle IlS 11 • a for nil a Ilnd b. In faet, it can be seen in the above example that 1 • 2 = 2, whereas 2. 1 = a. One must consequently take care to refer to a • b as the product. of a and b and to b • a as the product of b and a; the distinction is quite importallt.

Ive gnlUp. Examp'e 2-21. As 1\ silllple exampl(! of a noneollllllutative group, let the set G consist of all ordered pail'R of real numbers with nonzero first component: G= {(a,b)la,bER',a~O}. Define the operation • on G by the formula (a, b) • (e, d) = (ae, be + Il). he oJlI~mt,ioll flllluWH frolll tho f,ulli/inr properties of the real numbers, for wc have [(a, b) • (e, tl») • (e,f) + = (ae, be (l) • (e,J) = «ae)e, (be d)e f) = (a(ee), b(ee) (de f) = (a, b) • (ee, de + f) = (a, b) • [(e, d) • (e, f)].

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Algebraic Numbers and Algebraic Functions by Cohn P.M.

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