SELECTED CHAPTERS FROM ALGEBRA I. R. Shafarevich Preface

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10 I. R. <strong>Shafarevich</strong>approaches studying and research with such ease, uency, eagerness and peace as oilows form a pot and I wonder how canoneachieve somuch at that age". Furtheron, Theaetetus himself informs Socrates about the work he did with a friend, alsocalled Socrates, a namesake of the philosopher. He says that Theodor informedthem about the incommensurability (to use contemporary terms) of the side of asquare with the unit segment ifthe area of the square is an integer, but not thesquare of an integer. If this area is n, this means that p n is irrational. Theodorproved this for n = 2, 3, 5, and \solving one case after the other, he came up to 17".Theaetetus became intersted in the problem and, together with his friend Socrates,solved it, as it is recorded at the end of the dialogue. We shall not go into thereasoning of Theodor (there are several hypotheses), but we shall give the proof ofthe general statement, following the exposition of Euclid which is,very probablyanalogous to the proof of Theaetetus (with a simplication given by Gauss 2000years later).We rst prove an analog of Lemmas 1 and 3.THEOREM 4. For any two positive integers n and m there exist integers tand r, positive or 0, suchthatr n, the equality (4)holds with t = 0, r = n. If n > m, then put n 1 = n ; m. Clearly, n 1 > 0. Ifn 1 > m, putn 2 = n 1 ;m. We keep on subtracting m until we arrive atthenumbern t = n ; m ;;m = r, where r > 0 but t 2 . Subtracting the second equality from the rst we get:m(t 1 ; t 2 )+r 1 ; r 2 =0,i.e.m(t 1 ; t 2 )=r 2 ; r 1 . Since r 1
Selected chapters from algebra 11be prime (and hence a prime divisor of n) or it has two factors: a = a 1 b 1 . Thenn = a 1 (b 1 b)wherea 1
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Page 40 and 41: 18 I. R. Shafarevichall binomial co
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72 I. R. Shafarevichfollows: if M =
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74 I. R. Shafarevichis v(M 2 l) see
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76 I. R. ShafarevichIn order to exp
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78 I. R. ShafarevichApplying the sa
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80 I. R. Shafarevich7. Find the num
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16 I. R. Shafarevichn(M 1 \ M 2 )+n
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18 I. R. ShafarevichThis formula co
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20 I. R. ShafarevichWe can now appl
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22 I. R. Shafarevichthat n(M i1 \\M
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24 I. R. Shafarevich6. Let M be a n
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26 I. R. Shafarevichwhere on the ho
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28 I. R. Shafarevichthey do not inc
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30 I. R. ShafarevichThe experiment
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32 I. R. ShafarevichFor example, fo
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34 I. R. ShafarevichFig. 9introduce
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36 I. R. Shafarevichthe probability
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38 I. R. ShafarevichSet t =1into (1
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40 I. R. Shafarevichbility will be
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64 I. R. Shafarevichin x2 of Chapte
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66 I. R. ShafarevichProblems1. Prov
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68 I. R. Shafarevichis greater than
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70 I. R. Shafarevichexpression (n +
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72 I. R. Shafarevichelaborate metho
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74 I. R. ShafarevichNote that if we
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76 I. R. Shafarevicha log a x = x,
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78 I. R. Shafarevichalso divisible
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80 I. R. ShafarevichFrom here, usin
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82 I. R. Shafarevich6. Using the re
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2 I. R. Shafarevichsomewhere, and w
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4 I. R. ShafarevichVII(axiomofembed
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6 I. R. ShafarevichWe met in Chapte
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8 I. R. Shafarevichc n =1,thena n =
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10 I. R. Shafarevich4. Does there e
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12 I. R. ShafarevichTHEOREM 2. Two
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14 I. R. Shafarevichmust be satised
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16 I. R. Shafarevichnumber. From sc
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18 I. R. Shafarevich0 6 c ; a n 6 b
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20 I. R. Shafarevichjxj n;k > nja k
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22 I. R. Shafarevichsomewhere insid
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24 I. R. Shafarevichthat for x larg
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26 I. R. ShafarevichFor example, th
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28 I. R. ShafarevichThus, the chara
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30 I. R. Shafarevichand the formula
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32 I. R. Shafarevichthe sequence f
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34 I. R. ShafarevichConsider, for e
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SELECTED CHAPTERS FROM ALGEBRA I. R. Shafarevich Preface

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