SELECTED CHAPTERS FROM ALGEBRA I. R. Shafarevich Preface

Selected chapters from algebra 25Problems1. Notice that the area s n of the polygon which we calculated at the end of thesection is less than the area of the given gure, bounded by the parabola y = x 2 ,since the polygon is situated inside that gure. Construct the polygon made upfrom rectangles whose bases are segments from i i +1to and whose heights are 2 n ni +1which contains the given gure. Its area s 0 n will therefore be greater thannthe area of the gure. Calculate the area s 0 n and prove that as n increases, itapproaches 1=3. This gives a more convincing (i.e. more \strict") proof of the factthat the required area is 1=3.2. Try to solve the analogous problem for the \m-th degree parabola", givenby the equation y = x m . Verify that in order to obtain the result it is not necessaryto know the Bernoulli's polynomials g m (x) completely, but that is enough to knowthe coecient of the leading term a m+1 x m+1 . Prove that a m+1 = 1 and hencem +1nd the area of the gure bounded by the parabola whose equation is y = x m ,bythe x-axis and by the line x =1.3. Prove that the area of the gure bounded by the parabola y = x m ,x-axis and the line x = a is equal tothe polynomial1m +1 am+1 . Notice that the derivative of1m +1 xm+1 is x m . This is indeed an instance of Barrow's theoremthat integration and nding derivatives are operations inverse to each other.4. Prove that the sum of the binomial coecients with even upper indicesC 0 n + C 2 n + and with odd indices C 1 n + C 3 n + are equal and nd their mutualvalue.5. Find the relation between binomial coecients which expresses that(1 + x) n (1 + x) m = (1 + x) n+m . For n = m deduce the formula for the sumof the squares of binomial coecients.6. If p is a prime number, prove that all binomial coecients C k p for k 6= 0p,are divisible by p. Deduce from this that 2 p ; 2 is divisible by p. Prove thatforany integer n, thenumber n p ; n is divisible by p. This theorem was rst provedby Fermat.7. What can be said about the sequence a if all the terms of the sequence aare equal? What does formula (31) give in this case?8. Find the sum S 3 (n) and verify that S 3 (n) =(S 1 (n)) 2 .9. Let a be any sequence a 0 , a 1 , a 2 , ... Apply the operation once more tothe sequence a. The obtained sequence (a) will be denoted by 2 a. Dene k a by induction as ( k;1 a). When can we solve the so-called \innite interpolationproblem", that is to say when is there a polynomial f(x) of degree notgreater than m such thatf(n) =a n for n =0 1 2 ...? Prove that a necessary andsucient condition is given by( m+1 a) n = 0 for n > m. This condition shows thatif we write the sequence a, and under it the sequence whose terms are dierences