By A. I. Kostrikin, I. R. Shafarevich

ISBN-10: 0387170065

ISBN-13: 9780387170060

From the experiences: "... this is often one of many few mathematical books, the reviewer has learn from conceal to hide ...The major advantage is that just about on each web page you can find a few unforeseen insights... " Zentralblatt für Mathematik "... There are few proofs in complete, yet there's an exciting blend of sureness of foot and lightness of contact within the exposition... which transports the reader without problems around the entire spectrum of algebra...Shafarevich's e-book - which reads as conveniently as a longer essay - breathes existence into the skeleton and may be of curiosity to many periods of readers; definitely starting postgraduate scholars may achieve a most useful point of view from it but... either the adventurous undergraduate and the verified specialist mathematician will discover a lot to enjoy..."

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**Example text**

1) holds. 3. Explain why f (x) := |3 + 7|x|| − |3x2 − 8|9 5 − |1 − x2 | is continuous at 2. 4. Let f : R → R be determined by f (x) := 1 if x ≥ 3 . Then f is −1 if x < 3 discontinuous, that is not continuous, at 3. Proof. Since lim f (x) = −1 = 1 = lim f (x) x 3 x 3 f does not have a limit at 3, in particular, f is not continuous at 3. 5. Let f : C → R be determined by f (x) := 8 if x = 5 . Then 2 if x = 5 lim f (x) = 8 = 2 = f (5), x→5 hence f is discontinuous at 5. 6. Let a be some irrational number.

For a set A let 1A (x) := 1 if x ∈ A . 0 if x ∈ /A The function 1A is called the characteristic function of A. 21 (Dirichlet Function). Let f : R → R be the characteristic function of the set of rational, f := 1Q . Let a be a real number. By density of the rationals and of the irrationals a is an accumulation point of Q and of R \ Q. Clearly, lim f x→a lim f x→a Q (x) = lim 1 = 1, and x→a (x) = lim 0 = 0. 20 with D1 := Q and D2 := R \ Q, shows that f does not have a limit at the point a. Since a was arbitrary, f does not have a limit at any point.

If (xn ) is bounded and (yn ) is null, then (xn yn ) is null. 2. If (yn ) is null and |xn | ≤ |yn |, then (xn ) is null. 3. If xn ≤ yn and xn → ∞, then yn → ∞. 4. If zn = 0 for all n, then (zn ) is null iff 1/|zn | → ∞. 5. Find two sequences (xn ) and (yn ) such that xn → ∞, yn → 0 and xn yn → 1. 6. Find two sequences (xn ) and (yn ) such that xn → ∞, yn → 0 and xn yn → 29. 7. Find two sequences (xn ) and (yn ) such that xn → ∞, yn → ∞ and xn − yn → 1. Problems for Sect. 7 1. Let x ∈ R. Suppose 1 < x.

### Algebra I. Basic notions of algebra by A. I. Kostrikin, I. R. Shafarevich

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