By Thomas W. Hungerford

- summary ALGEBRA: AN creation is meant for a primary undergraduate path in glossy summary algebra. Its versatile layout makes it compatible for classes of varied lengths and assorted degrees of mathematical sophistication, starting from a conventional summary algebra path to 1 with a extra utilized flavor. The e-book is prepared round subject matters: mathematics and congruence. every one subject is built first for the integers, then for polynomials, and at last for earrings and teams, so scholars can see where many summary ideas come from, why they're very important, and the way they relate to at least one another.
*New Features:* - A
**groups-first alternative that permits those that are looking to hide teams ahead of earrings to take action easily.** **Proofs for beginners**within the early chapters, that are damaged into steps, every one of that's defined and proved in detail.- In the center path (chapters 1-8), there are
**35% extra examples**and**thirteen% extra exercises.**

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**Extra info for Abstract algebra : an introduction**

**Sample text**

Hence D3 attains its minimum value of zero at the corners ofT, the points (0, 0), (0, 1), (1, 0), which proves (J 3 ). Now it is clear that this argument easily extends to the general case. REMARK (i) D Of course D2(s) is defined for all s E JR for which (1- s)x + sy E I and the argument given above shows that D~ is strictly decreasing for all such s. This implies that D 2 is negative outside [0, 1]; that is == 0 if s == 0 1· D2 (s) { > 0 if 0 < s ' <' 1; < 0 if s < 0 or 1 < s. So in the case n == 2, ( J2) only holds for positive weights.

THEOREM 3 If x > 1, and 0 < r < 1, or if x < 1 and r > 1 then (x- 1) 2 1 x2 ( 1 - r) 2 If x 0 < xr - 1 r (x - 1) - < 1 and 0 < r < 1 or if x > 1 and r > 1 < 1 ( 1 - r) x ( x - 1) 2 . 2 (4) then (rv4) holds . These inequalities follow using a calculus argument similar to that used to 0 prove (B). COROLLARY 4 (1 + x)r ( 1 + 0 (r 2 )) 0 r---+ 11 r + rx + O(x 2 ), as x---+ 0; == 1 + 0 (r), as r ---+ 0. ] D REMARK (viii) An alternative sharpening of (B) can be found in [Haber]. 81-83] If X# (a) THE ExPo- e then (7) To see this note that y == x / e is a tangent to the curve y == log x at the point (e, 1).

3 Example(i) and that in Notations 7, lim pk == ~ J. Details can be found in the reference. 0 n Taking p == e and f(a) == 2:~ 1 g(ai), where g is convex this theorem gives a proof of the equal weight case of (J). k~oo As has been suggested (J) includes many of the classical inequalities as special cases, and many other inequalities are often a much disguised form of (J); see [He J K]. For instance we have the following. COROLLARY 14 If w is a positive n-tuple then n w~n >IT w;:i. i=l D Use the concavity of the logarithmic function.